# SAT Exponents

To solve problems with exponents on the SAT, you need to know how to:

- apply the rules of exponents
- manipulate algebraic expressions
- solve simple equations
- solve exponential equations

#### Contents

## Examples

If \( a^2 = 10 \), what is the value of \( a^4 \)?

(A) \( \ \ 1 \)

(B) \( \ \ \sqrt{10} \)

(C) \( \ \ 10 \)

(D) \(\ \ 100 \)

(E) \( \ \ 10000 \)

Correct Answer: D

Solution 1:

Tip: Know the Rules of Exponents.

\( a^4= a ^ { 2 + 2 } = a^2 \times a^2 = 10 \times 10 = 100 \).

Solution 2:

Tip: Know the Rules of Exponents.

Since \( a ^2 = 10 \), so \( a =\sqrt{10}\). Then, \( a^4 = (\sqrt{10})^4 = \left(10 ^ \frac{1}{2} \right) ^ 4 = 10 ^ { ( \frac{1}{2} \times 4)} = 10 ^ 2 = 100 \).

Incorrect Choices:

(A)

Tip: The simplest choice may not be the correct one.

(B)

Tip: Read the entire question carefully.

If you solve for \( a \), you will obtain \( a = \sqrt{10} \). However, the question asks for \( a^4 .\)

(C)

Tip: Just because a number appears in the question doesn't mean it is the answer.

(E)

Tip: Read the entire question carefully.

If you forget the exponent and think that \( a ^ { \boxed{1} } = 10 \), you will obtain \( a^4 = 10^4 = 10000 \).

If \(a^{2x} \cdot a^{3} = a^{15}\) and \(\sqrt[3]{b^{y}} = b^{5}\), what is the value of \(y-x\)?

(A) \(\ \ -1\)

(B) \(\ \ 6\)

(C) \(\ \ 9\)

(D) \(\ \ \frac{25}{2}\)

(E) \(\ \ 21\)

Correct Answer: C

Solution:

Tip: Know the Rules of Exponents.

Using the first equation, we solve for \(x\).\[\begin{array}{l l c l} a^{2x} \cdot a^{3} &=& a^{15} &\quad \text{first equation}\\ a^{2x+3} &=& a^{15} &\quad \text{apply}\ a^{m} \cdot a^{n} = a^{m+n}\ \text{to left side}\\ 2x+3 &=& 15 &\quad \text{equate exponents}\\ 2x &=& 12 &\quad \text{subtract}\ 3\ \text{from both sides}\\ x &=& 6 &\quad \text{divide both sides by}\ 2\\ \end{array}\]

Similarly, using the second equation, we solve for \(y\).

\[\begin{array}{l l c l} \sqrt[3]{b^{y}} &=& b^{5} &\quad \text{second equation}\\ b^{\frac{y}{3}} &=& b^{5} &\quad \text{apply}\ \sqrt[n]{a^{m}}=a^{\frac{m}{n}}\ \text{to left side}\\ \frac{y}{3} &=& 5 &\quad \text{equate exponents}\\ y &=& 15 &\quad \text{multiply both sides by}\ 3\\ \end{array}\]

Now that we know \(x=6\) and \(y=15\), we find \(y-x=15-6=9\).

Incorrect Choices:

(A)

Tip: Read the entire question carefully.

If you ignore the cube root in the second equation and solve \(b^{y}=b^{5}\) instead of \(\sqrt[3]{b^{y}} = b^{5}\), you will get \(y=5\). Then \(y-x=5-6=-1\). But this is wrong.

(B)

Tip: Read the entire question carefully.

If you solve for \(x\), instead of \(y-x\), you will get this wrong answer.

(D)

Tip: Know the Rules of Exponents.

If you apply the wrong formula \(a^{m} \cdot a^{n} = a^{\boxed{m \cdot n}}\) to the first equation, like this:\[\begin{array}{l l c l} a^{2x} \cdot a^{3} &=& a^{15} &\quad \text{first equation}\\ \boxed{a^{2x\cdot 3}}&=& a^{15} &\quad \text{mistake: multiplied exponents, instead of added}\\ \end{array}\]

you will get \(x=\frac{5}{2}\). Then \(y-x=15-\frac{5}{2}=\frac{25}{2}\). The correct rule states \(a^{m} \cdot a^{n} = a^{m + n}\).

(E)

Tip: Read the entire question carefully.

If you solve for \(y+x\) instead of \(y-x\), you will get this wrong answer.

The expression \(\frac{(3y^{3})(2y^{2})^{2}}{(y^{4})^{3}} \cdot (y^{0})^{3}\) is equivalent to which one of the following?

(A) \(\ \ 12y^{-5}\)

(B) \(\ \ 6y^{-5}\)

(C) \(\ \ 12y^{-2}\)

(D) \(\ \ 12\)

(E) \(\ \ 12y^{5}\)

Correct Answer: A

Solution:

Tip: Know the Rules of Exponents.

Using the rules of exponents, we simplify the given expression.\[\begin{array}{l c l l l} \frac{(3y^{3})(2y^{2})^{2}}{(y^{4})^{3}} \cdot (y^{0})^{3} &=& \frac{(3y^{3})(2y^{2})^{2}}{(y^{4})^{3}} \cdot (1)^{3} &\quad \text{apply}\ a^{0}=1\ &(1)\\ &=& \frac{(3y^{3})(2y^{2})^{2}}{(y^{4})^{3}} &\quad \text{apply}\ 1^{a}=1 &(2)\\ &=& \frac{(3y^{3})(2^{2}y^{2\cdot 2})}{(y^{4})^{3}} &\quad \text{apply}\ (a \cdot b)^{n}=a^{n} \cdot b^{n}\ &(3)\\ &=& \frac{(3 y^{3}) \cdot 4 y^{4}}{(y^{4})^{3}} &\quad \text{simplify numerator}\ &(4)\\ &=& \frac{12 \cdot y^{3} \cdot y^{4}}{(y^{4})^{3}} &\quad\ 3 \cdot 4=12\ &(5)\\ &=& \frac{12 \cdot y^{7}}{(y^{4})^{3}} &\quad \text{apply}\ a^{m}a^{n}=a^{m+n}\ &(6)\\ &=& \frac{12 \cdot y^{7}}{y^{4 \cdot 3}} &\quad \text{apply}\ (a^{n})^{m}=a^{n \cdot m}\ &(7)\\ &=& \frac{12 \cdot y^{7}}{y^{12}} &\quad \text{simplify}\ &(8)\\ &=& 12y^{-5} &\quad \text{apply}\ \frac{a^{m}}{a^{n}}=a^{m-n}\ &(9)\\ \end{array}\]

Incorrect Choices:

(B)

Tip: Know the Rules of Exponents.

If in step \((3)\) you don't apply the rule (a \(\cdot b)^{n}=a^{n} \cdot b^{n}\) correctly, you will get this wrong answer. That is,\[\begin{array}{r l} \frac{(3y^{3})(\fbox{2}y^{2\cdot 2})}{(y^{4})^{3}} &\quad \text{mistake: did not raise}\ 2\ \text{to the power of}\ 2\ &(3)\\ \end{array}\]

(C)

Tip: Read the entire question carefully.

If you forget the \(0\) in the multiplicand's exponent and read \(\frac{(3y^{3})(2y^{2})^{2}}{(y^{4})^{3}} \cdot (y)^{3}\) instead of \(\frac{(3y^{3})(2y^{2})^{2}}{(y^{4})^{3}} \cdot (y^{\boxed{0}})^{3}\), you will get this wrong answer.

(D)

Tip: Know the Rules of Exponents.

If in step \((7)\) you apply the wrong rule \((a^{n})^{m}=a^{\fbox{n + m}}\) to the denominator, as shown below, you will get this wrong answer. Don't add the exponents when you should be multiplying them.\[\begin{array}{r l} \frac{12y^{7}}{y^{4 \fbox{+} 3}} &\quad \text{mistake: added exponents}\\ &\quad \text{instead of multiplied}\ &(7)\\ \end{array}\]

The correct rule states \((a^{n})^{m}=a^{n \cdot m}\).

(E)

Tip: Know the Rules of Exponents.

Tip: Be careful with signs!

If in the step \((9)\) you apply the wrong rule \(\frac{a^{m}}{a^{n}}=a^{\fbox{n-m}}\), you will get this wrong answer. The correct rule states \(\frac{a^{m}}{a^{n}}=a^{m-n}\).Alternatively, you may not notice or you may forget that the exponent becomes negative. Be careful!

## Review

If you thought these examples difficult and you need to review the material, these links will help:

- Rules of Exponents

\( a^m \times a^n = a^{ m + n } \), \( a^n / a^m = a^ { n - m }\ldots\) - Solving Exponential Equations
- Rules of Exponents applied to numbers
- Multi-Step Equations-Basic

## SAT Tips for Exponents

- Know the Rules of Exponents.
- Recognize first few perfect squares \((1, 4, 9, ... 400)\) and cubes \((1, 8, 27,... 1000).\)
- Be careful with signs!
- Follow order of operations.
- SAT General Tips