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

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!

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

• 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