SAT Exponents

Correct Answer: C

Solution 1:

Tip: Know the Rules of Exponents.
Tip: Recognize first few perfect squares $(1, 4, 9, ... 400)$ and cubes $(1, 8, 27,... 1000).$
We begin with the first equation. Since the unknowns are in the exponents, we will manipulate both sides of the equation so that they have the same base. Then we will equate the exponents and solve for the unknowns. We notice that the left hand side is the product of powers of $4$ . If you can't tell that the right hand side is also a power of $4$ , use your calculator to check.

$\begin{array}{l c l l} 4^{a} \cdot 4^{b} &=& 1024 &\quad \text{given}\\ 4^{a+b} &=& 4^{5} &\quad \text{apply}\ a^{m} \cdot a^{n} = a^{m+n}\ \text{to left side,}\\ &&& \quad \text{and notice that}\ 1024=4^{5}\\ \end{array}$

Now that the bases are the same, we can equate the exponents. We find that:

$\begin{array}{l c l l} a+b &=& 5 &\quad \text{equate the exponents}\\ \end{array}$

The only two positive non-consecutive integers whose sum is $5$ are $1$ and $4$ . Let $a=1$ and $b=4$ . Then, $2^{a} + 2^{b} = 2^{1}+2^{4}=2+16=18$ .

Note: If we let $a=4$ and $b=1$ , we'll get the same result.

Solution 2:

Tip: Know the Rules of Exponents.
Tip: Recognize first few perfect squares $(1, 4, 9, ... 400)$ and cubes $(1, 8, 27,... 1000).$
Here we manipulate both sides of the equation so that each has a base of $2$ . We use $4=2^{2}$ and $1024=2^{10}$ . If you can't tell immediately that $1024$ is a power of $2$ , use your calculator to check.

We begin with the first equation:

$\begin{array}{l c l l} 4^{a} \cdot 4^{b} &=& 1024 &\quad \text{given}\\ (2^{2})^{a} \cdot (2^{2})^{b} &=& 2^{10} &\quad \text{notice that}\ 4=2^{2}\ \text{and}\ 1024=2^{10}\\ 2^{2a} \cdot 2^{2b} &=& 2^{10} &\quad \text{apply}\ (a^{m})^{n}=a^{mn}\ \text{to left side}\\ 2^{2a+2b} &=& 2^{10} &\quad \text{apply}\ a^{m} \cdot a^{n} = a^{m+n}\ \text{to left side}\\ \end{array}$

Now that the bases are the same, we can equate the exponents. We find that:

$\begin{array}{l c l l} 2a+2b &=& 10 &\quad \text{equate the exponents}\\ a+b &=& 5 &\quad \text{divide both sides by}\ 2\\ \end{array}$

The only two positive non-consecutive integers whose sum is $5$ are $1$ and $4$ . Let $a=1$ and $b=4$ . Then, $2^{a} + 2^{b} = 2^{1}+2^{4}=2+16=18$ .

Note: If we let $a=4$ and $b=1$ , we'll get the same result.

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Incorrect Choices:

(A)
Tip: Read the entire question carefully.
If you solve for $a+b$ instead of $2^{a} + 2^{b}$ , you will get this wrong answer.

(B)
Tip: Read the entire question carefully.
If you select the consecutive integers $2$ and $3$ as $a$ and $b$ , you will get this wrong answer. The question asks for two positive non-consecutive integers.

(D)
Tip: Know the Rules of Exponents.
You may get this wrong answer if in the final step, you make a mistake when you apply the rule $a^{m} \cdot a^{n} = a^{m+n}$ , like this:

$2^{a} + 2^{b} = \boxed{2^{1}+2^{4} = 2^{1+4}} = 2^{5} = 32$ .

(E)
Tip: Know the Rules of Exponents.
Tip: Just because a number appears in the question doesn’t mean it is the answer.
Suppose that, using the first equation, you find that $4^{a+b} = 1024$ . Suppose then that you make a mistake when you apply the rule $a^{m} \cdot a^{n} = a^{m+n}$ to the second equation, as shown below. You may conclude that the desired quantity, $2^{a}+2^{b}$ equals

$\boxed{2^{a} + 2^{b} = (2 \cdot 2)^{a + b}} = 4^{a+b} = 1024$

or,

$\boxed{2^{a} + 2^{b} =(2 + 2)^{a + b}} = 4^{a+b} = 1024$ ,

but you will be wrong.

Alternatively, you may select this answer just because it appears in the prompt, but you will be wrong.

awesome explanation