12. Algebraic Manipulations · Mastering AMC 10/12

12.00%

Exponent Rules

In this section you will be able to use the exponent rules to turn equations between powers into equations between exponents.

$2^3 \cdot 2^5$ is three 2’s times five more 2’s: eight 2’s in all, so it equals $2^8$ . Every exponent rule is just counting factors like this.

The book prints these rules before Section 12.1, so they get their own short lesson here.

When both sides of an equation are powers of the same base, the exponents must be equal. Look: that turns a scary equation like $3^a = 81^b$ into an ordinary linear one: $81 = 3^4$ , so $a = 4b$ .

Exploreplay first, formulas later

Try this: Select both rows and press Multiply. Which exponent rule turns 3 and 5 into 8?
Try this: Select x⁵ = 32 first, then x³ = 8, and press Divide. What is x², and what is x?
Try this: Select row 1, set the power to −1 and press Raise. Why is x⁻³ equal to 1/8?

Name it

Theorem 12.0.1 · Exponent Rules

For any nonzero $x$ :

x^{-a} = \frac{1}{x^{a}}

x^{a} \times x^{b} = x^{a+b}

x^{a} \div x^{b} = x^{a-b}

\left(x^{a}\right)^{b} = x^{ab}

Try it

Problem 12.6.7

AMC 10

For what value $x$ does $10^{x} \cdot 100^{2x} = 1000^{5}$ ?

Problem 12.6.3

AMC 10

Real numbers $a$ and $b$ satisfy the equations $3^a = 81^{b+2}$ and $125^b = 5^{a-3}$ . What is $ab$ ?

Contest cornerreal contest problems

Problem 12.6.19(additional)

AIME

The equation $2^{333x-2} + 2^{111x+2} = 2^{222x+1} + 1$ has three real roots. Given that their sum is $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m + n$ .

Checkpoint for section 12.0

Get 2 of 3 right to master this section. One try each, no hints.

12.10%

Binomial Expansions

In this section you will be able to expand squares and cubes of sums, and climb from x + 1/x to higher powers without finding x.

If $x + 1/x = 4$ , can you find $x^4 + 1/x^4$ without ever finding $x$ ? Yes: just square twice.

A square of a sum is an area: the $(x+y)$ by $(x+y)$ square splits into $x^2$ , two $xy$ rectangles and $y^2$ .

Look for sums and products. If a problem gives you $x + y$ and $xy$ , then $x^2 + y^2$ , $(x - y)^2$ and $x^3 + y^3$ all follow, with no square roots needed.

The reciprocal pair $x$ and $1/x$ is special because their product is $1$ . Every time you square $x^n + 1/x^n$ , the cross term is just $2$ .

Exploreplay first, formulas later

Try this: Tap one xy tile. How many xy tiles are there? That is where the 2xy comes from.
Try this: Switch to 4xy and press Take out the middle square. What area is left over?
Try this: Switch to (x+y)³ and count the blocks of each color. Where do the two 3s come from?

Name it

Theorem 12.1.1 · Binomial Square Expansions

(x + y)^2 = x^2 + 2xy + y^2 = (x - y)^2 + 4xy

(x - y)^2 = x^2 - 2xy + y^2 = (x + y)^2 - 4xy

(x + y)^2 + (x - y)^2 = 2\left(x^2 + y^2\right)

(x + y)^2 - (x - y)^2 = 4xy

(x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + xz)

Theorem 12.1.2 · Binomial Cube Expansions

(x + y)^3 = x^3 + 3xy(x + y) + y^3

(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3

(x - y)^3 = x^3 - 3xy(x - y) - y^3

(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3

x^3 + y^3 + z^3 - 3xyz = (x + y + z)\left(x^2 + y^2 + z^2 - xy - xz - yz\right)

Theorem 12.1.3

If

x + \frac{1}{x} = a

then

x^2 + \frac{1}{x^2} = a^2 - 2

x^3 + \frac{1}{x^3} = a^3 - 3a

x^4 + \frac{1}{x^4} = \left(a^2 - 2\right)^2 - 2

Watch itpredict, then reveal

Example 12.1

AMC 10

Suppose that the number $a$ satisfies the equation $4 = a + a^{-1}$ . What is the value of $a^4 + a^{-4}$ ?

1
You could solve for $a$ , but $a = 2 \pm \sqrt{3}$ is messy. Instead, square both sides of $a + \frac{1}{a} = 4$ . What is $a^2 + \frac{1}{a^2}$ ?

Predict first:
Your prediction

open on YouTube

Example 12.2

Given that $x - \frac{1}{x} = 3$ , find $x^3 - \frac{1}{x^3}$ .

1
Which identity from Theorem 12.1.2 has a cube of a difference?

Predict first:
Click a choice or type its letter (A–E).

open on YouTube

Try it

Problem 12.6.11

AMC 10

Real numbers $x$ and $y$ satisfy the equation $x^2 + y^2 = 10x - 6y - 34$ . What is $x + y$ ?

Problem 12.6.12(additional)

If $x + \frac{1}{x} = 4$ , then what is the value of $x^8 + \frac{1}{x^8}$ ?

Contest cornerreal contest problems

Problem 12.6.6

AMC 10

Real numbers $x$ and $y$ satisfy $x + y = 4$ and $x \cdot y = -2$ . What is the value of

x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y?

Problem 12.6.24(additional)

AIMEChallenge

Let $x$ and $y$ be real numbers satisfying $x^4y^5 + y^4x^5 = 810$ and $x^3y^6 + y^3x^6 = 945$ . Evaluate $2x^3 + (xy)^3 + 2y^3$ .

Checkpoint for section 12.1

Get 2 of 3 right to master this section. One try each, no hints.

12.20%

Quadratic Factorizations

In this section you will be able to spot a difference of squares and factor it, even when the squares are hiding inside square roots.

√(49 − x²) − √(25 − x²) = 3. Multiply it by its partner √(49 − x²) + √(25 − x²), and the square roots vanish.

A difference of squares always factors: $x^2 - y^2 = (x - y)(x + y)$ . Read it in both directions.

Left to right, it breaks a number or expression into a product. Right to left, multiplying a difference by the matching sum kills the cross terms. That is how you clear square roots.

Exploreplay first, formulas later

Try this: Press Cut and move the strip. What are the side lengths of the new rectangle?
Try this: Here a = 7 and b = 5. Why is 49 − 25 the same as 2 × 12?
Try this: In Problem 12.6.9, the two square roots play the roles of a and b. You know a − b. What do you know about a² − b²?

Name it

Theorem 12.2.1 · Difference of Squares

x^2 - y^2 = (x - y)(x + y)

Try it

Problem 12.6.9

AMC 10

Suppose that real number $x$ satisfies

\sqrt{49 - x^2} - \sqrt{25 - x^2} = 3.

What is the value of $\sqrt{49 - x^2} + \sqrt{25 - x^2}$ ?

Problem 12.6.8

AMC 12

Given that $x$ and $y$ are distinct nonzero real numbers such that $x + \frac{2}{x} = y + \frac{2}{y}$ , what is $xy$ ?

Contest cornerreal contest problems

Problem 12.6.5

AMC 10

If $y + 4 = (x - 2)^2$ , $x + 4 = (y - 2)^2$ , and $x \neq y$ , what is the value of $x^2 + y^2$ ?

Problem 12.6.14(additional)

AMC 10

Which of the following is equivalent to $\sqrt{\dfrac{x}{1 - \frac{x-1}{x}}}$ when $x < 0$ ?

Checkpoint for section 12.2

Get 2 of 3 right to master this section. One try each, no hints.

12.30%

Cubic Factorizations

In this section you will be able to factor sums and differences of cubes and use them on reciprocal pairs.

x − 1/x = 3. Find x³ − 1/x³ with one factorization.

Cubes factor too, but the second factor is a quadratic: $x^2 + xy + y^2$ for a difference, $x^2 - xy + y^2$ for a sum.

With $y = 1/x$ the middle term $xy$ is just $1$ . So $x^3 - 1/x^3$ splits into $x - 1/x$ times $x^2 + 1 + 1/x^2$ .

Exploreplay first, formulas later

Try this: Here a = 5 and b = 2. Check that 125 − 8 = 3 × 39 using the two factors.
Try this: Switch to a³ + b³. Which factor changed, and which sign flipped?
Try this: Set b = 1. Why is a³ − 1 always divisible by a − 1?

Name it

Theorem 12.3.1 · Difference of Cubes

x^3 - y^3 = (x - y)\left(x^2 + xy + y^2\right)

Theorem 12.3.2 · Sum of Cubes

x^3 + y^3 = (x + y)\left(x^2 - xy + y^2\right)

Watch itpredict, then reveal

Example 12.3

Given that $x - \frac{1}{x} = 3$ , find $x^3 - \frac{1}{x^3}$ .

1
This is Example 12.2 again. This time, factor $x^3 - \frac{1}{x^3}$ as a difference of cubes with $y = \frac{1}{x}$ . What do you get?

open on YouTube

Try it

Problem 12.6.13(additional)

MATHCOUNTS

If $\left(a - \frac{1}{a}\right)^2 = 4$ , what is the absolute value of $\left(a^3 - \frac{1}{a^3}\right)$ ?

Checkpoint for section 12.3

Get 2 of 3 right to master this section. One try each, no hints.

12.40%

Higher Power Factorizations

In this section you will be able to factor higher powers and use Simon's Favorite Factoring Trick on equations with xy, x and y terms.

xy + x + y = 71 looks stuck, until you add 1 to both sides.

$x^n - y^n$ always has a factor $x - y$ . For odd $n$ , $x^n + y^n$ always has a factor $x + y$ .

Simon’s Favorite Factoring Trick completes a rectangle: $xy + x + y$ is missing the corner piece $1$ . Add it, and the left side becomes $(x + 1)(y + 1)$ . With integers, that leaves only a few factor pairs to try.

Exploreplay first, formulas later

Try this: Press the Add button. What does xy + x + y + 1 factor into?
Try this: Which factor pairs of 72 give positive x and y?
Try this: Which of those pairs also fits x²y + xy² = 880?

Name it

Theorem 12.4.1 · nth Power Factorizations

Sum of odd powers:

x^{2n+1} + y^{2n+1} = (x + y)\left(x^{2n} - x^{2n-1}y + x^{2n-2}y^2 - \cdots - xy^{2n-1} + y^{2n}\right)

The signs in the second factor alternate between positive and negative.

Difference of powers:

x^{n} - y^{n} = (x - y)\left(x^{n-1} + x^{n-2}y + x^{n-3}y^2 + \cdots + xy^{n-2} + y^{n-1}\right)

The signs in the second factor are all positive.

Theorem 12.4.2 · Simon's Favorite Factoring Trick

xy + kx + jy + jk = (x + j)(y + k)

Watch itpredict, then reveal

Example 12.4

Omega Learn

If $x > y$ , $x + y = 8$ , and $xy = 4$ , the value of $x^4 - y^4$ is $a\sqrt{b}$ . Find $a + b$ .

1
Factor $x^4 - y^4$ as a difference of squares, and then factor again. What do you get?

open on YouTube

Try it

Problem 12.6.15(additional)

AIME

Find $x^2 + y^2$ if $x$ and $y$ are positive integers such that

xy + x + y = 71, \qquad x^2y + xy^2 = 880.

Contest cornerreal contest problems

Problem 12.6.25(additional)

AIMEChallenge

The equation $2000x^6 + 100x^5 + 10x^3 + x - 2 = 0$ has exactly two real roots, one of which is $\frac{m + \sqrt{n}}{r}$ , where $m$ , $n$ and $r$ are integers, $m$ and $r$ are relatively prime, and $r > 0$ . Find $m + n + r$ .

Checkpoint for section 12.4

Get 2 of 3 right to master this section. One try each, no hints.

12.50%

Algebraic Equations

In this section you will be able to combine equations by adding, subtracting, multiplying and substituting, and find exactly what the problem asks for.

a²b = 4, b²c = 3, c²a = 18. Solving for one letter at a time is painful. Multiply all three!

Contest systems are rarely solved one variable at a time. Look at the shape first: are the equations symmetric? Does one block repeat? Does the question ask for a combination, like $xyz$ or $a + b + c + d$ ?

Then pick the move that fits. Multiply when the unknowns appear as products, add or subtract when they appear as sums, and substitute when the same chunk shows up again and again.

Exploreplay first, formulas later

Try this: Select all three rows and press Multiply. What is (abc)³?
Try this: Take the cube root of the new row. Then divide row 1 by it. What is a/c?
Try this: Can you reach a⁹ = 512 using only powers, products and quotients of the three given rows?

Name it

Concept 12.5.1 · Algebraic Equations Techniques

Substitution
Elimination
Adding/Subtracting/Multiplying Equations
Group terms together
Find common denominator
Don't always have to find every variable
Use symmetry between variables
Make smart substitutions that can simplify your expression
Look for common factorizations

Watch itpredict, then reveal

Example 12.5

Omega Learn

Solve for $a$ in

a^2b = 4, \qquad b^2c = 3, \qquad c^2a = 18.

1
Each letter appears with total exponent $3$ across the three equations. Multiply all three. What is $(abc)^3$ ?

Predict first:
Your prediction

open on YouTube

Example 12.6

AMC 12

If $x$ , $y$ , and $z$ are positive numbers satisfying

x + \frac{1}{y} = 4, \qquad y + \frac{1}{z} = 1, \qquad \text{and} \qquad z + \frac{1}{x} = \frac{7}{3},

then what is the value of $xyz$ ?

1
We want a product, so multiply the three equations. What is the right side, $4 \cdot 1 \cdot \frac73$ ?

Predict first:
Your prediction

open on YouTube

Example 12.7

AIME

Find the positive solution to

\frac{1}{x^2 - 10x - 29} + \frac{1}{x^2 - 10x - 45} - \frac{2}{x^2 - 10x - 69} = 0

1
The block $x^2 - 10x$ shows up three times. Substitute $y = x^2 - 10x - 29$ . Write the second denominator, $x^2 - 10x - 45$ , in terms of $y$ .

Predict first:
Your prediction

open on YouTube

Example 12.8

AIME

Let $(a, b, c)$ be the real solution of the system of equations $x^3 - xyz = 2$ , $y^3 - xyz = 6$ , $z^3 - xyz = 20$ . The greatest possible value of $a^3 + b^3 + c^3$ can be written in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

1
Let $p = abc$ . Write each cube in terms of $p$ .

open on YouTube

Try it

Problem 12.6.1

AMC 10

Assuming $a \neq 3$ , $b \neq 4$ , and $c \neq 5$ , what is the value in simplest form of the following expression?

\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}

Problem 12.6.2

AMC 10

Suppose that $x$ and $y$ are nonzero real numbers such that $\frac{3x+y}{x-3y} = -2$ . What is the value of $\frac{x+3y}{3x-y}$ ?

Problem 12.6.4

AMC 10

Let $a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5$ . What is $a + b + c + d$ ?

Problem 12.6.10

AMC 10

Two non-zero real numbers, $a$ and $b$ , satisfy $ab = a - b$ . Which of the following is a possible value of $\frac{a}{b} + \frac{b}{a} - ab$ ?

Contest cornerreal contest problems

Problem 12.6.17(additional)

AIME

Suppose that $x$ , $y$ , and $z$ are three positive numbers that satisfy the equations $xyz = 1$ , $x + \frac{1}{z} = 5$ , and $y + \frac{1}{x} = 29$ . Then $z + \frac{1}{y} = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

Problem 12.6.18(additional)

AIME

Find $ax^5 + by^5$ if the real numbers $a$ , $b$ , $x$ , and $y$ satisfy the equations

\begin{aligned} ax + by &= 3, \\ ax^2 + by^2 &= 7, \\ ax^3 + by^3 &= 16, \\ ax^4 + by^4 &= 42. \end{aligned}

s_k = a x^k + b y^k \quad\Rightarrow\quad s_k = p\,s_{k-1} + q\,s_{k-2}

$s_{1} = ax^{1} + by^{1}$ $3$ given
$s_{2} = ax^{2} + by^{2}$ $7$ given
$s_{3} = ax^{3} + by^{3}$ $16$ given
$s_{4} = ax^{4} + by^{4}$ $42$ given
$s_{5} = ax^{5} + by^{5}$ ?

Goal:

s_{5} = ax^{5} + by^{5}

Fit sₖ = p·sₖ₋₁ + q·sₖ₋₂ to the four given values, then run it forward. (Written this way, p = x + y and q = −xy.)

Problem 12.6.20(additional)

AIME

What is the product of the real roots of the equation $x^2 + 18x + 30 = 2\sqrt{x^2 + 18x + 45}$ ?

Checkpoint for section 12.5

Get 2 of 3 right to master this section. One try each, no hints.

12.6

Practice Problems

In this section you will be able to mix every manipulation from this chapter on harder contest problems.

Time to mix it all up. These AIME problems need bounding, clever substitutions, grouping and symmetry, often all at once. Every problem has three hints if you get stuck.

Want every problem from this chapter on one page? Open the problem set.

Contest cornerreal contest problems

Problem 12.6.16(additional)

AIME

Let $x_1 \leq x_2 \leq \cdots \leq x_{100}$ be real numbers such that $|x_1| + |x_2| + \cdots + |x_{100}| = 1$ and $x_1 + x_2 + \cdots + x_{100} = 0$ . Among all such 100-tuples of numbers, the greatest value that $x_{76} - x_{16}$ can achieve is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

Problem 12.6.21(additional)

AIME

Find $A^2$ , where $A$ is the sum of the absolute values of all roots of the following equation:

x = \sqrt{19} + \cfrac{91}{\sqrt{19} + \cfrac{91}{\sqrt{19} + \cfrac{91}{\sqrt{19} + \cfrac{91}{\sqrt{19} + \cfrac{91}{x}}}}}

Problem 12.6.22(additional)

AIMEChallenge

Let $a$ , $b$ , $c$ , and $d$ be real numbers that satisfy the system of equations

\begin{aligned} a + b &= -3, \\ ab + bc + ca &= -4, \\ abc + bcd + cda + dab &= 14, \\ abcd &= 30. \end{aligned}

There exist relatively prime positive integers $m$ and $n$ such that

a^2 + b^2 + c^2 + d^2 = \frac{m}{n}.

Find $m + n$ .

Problem 12.6.23(additional)

AIMEChallenge

Let $m$ be the largest real solution to the equation

\frac{3}{x-3} + \frac{5}{x-5} + \frac{17}{x-17} + \frac{19}{x-19} = x^2 - 11x - 4.

There are positive integers $a$ , $b$ , and $c$ such that $m = a + \sqrt{b + \sqrt{c}}$ . Find $a + b + c$ .