Exponential Operations - SAT Math

Card 0 of 20

Question

If $y = 5^{a} * 5^{b}$ and , what is the value of ?

Answer

Multiplying two exponents that have the same base is the equivalent of simply adding the exponents.

So $y = 5^{a} * 5^{b}$ is the same as $y = 5^{a + b}$ , and if , then $y = 5^{3}$ or .

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Question

If $3^{4x+6}=27^{2x}$ , what is the value of ?

Answer

Using exponents, 27 is equal to 33. So, the equation can be rewritten:

34_x_ + 6 = (33)2_x_

34_x_ + 6 = 36_x_

When both side of an equation have the same base, the exponents must be equal. Thus:

4_x_ + 6 = 6_x_

6 = 2_x_

x = 3

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Question

If _a_2 = 35 and _b_2 = 52 then _a_4 + _b_6 = ?

Answer

_a_4 = _a_2 * _a_2 and _b_6= _b_2 * _b_2 * _b_2

Therefore _a_4 + _b_6 = 35 * 35 + 52 * 52 * 52 = 1,225 + 140,608 = 141,833

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Question

If $9^{(x + 5)}+3^{2(x+5)}=162$ , what is the value of ?

Answer

Since we have two ’s in $9^{(x + 5)} + 3^{2(x + 5)}$ we will need to combine the two terms.

For $3^{2(x + 5) }$ this can be rewritten as

$(3^2)^{ (x + 5)} = 9^ {(x + 5)}$

So we have $9^{ (x + 5) }+ 9^{ (x + 5)} = 162$ .

Or $2 (9^{ (x + 5)}) = 162$

Divide this by : $9^{ (x + 5)} = 81 = 9^ 2$

Thus or

*Hint: If you are really unsure, you could have plugged in the numbers and found that the first choice worked in the equation.

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Question

If $5^2 \times 5^n = 5^{12}$ , what is the value of ?

Answer

Since the base is 5 for each term, we can say 2 + n =12. Solve the equation for n by subtracting 2 from both sides to get n = 10.

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Question

Simplify: y3x4(yx3 + y2x2 + y15 + x22)

Answer

When you multiply exponents, you add the common bases:

y4 x7 + y5x6 + y18x4 + y3x26

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Question

Solve for x.

23 + 2x+1 = 72

Answer

The answer is 5.

8 + 2x+1 = 72

2x+1 = 64

2x+1 = 26

x + 1 = 6

x = 5

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Question

What is the value of such that ?

Answer

We can solve by converting all terms to a base of two. 4, 16, and 32 can all be expressed in terms of 2 to a standard exponent value.

$4=2^2\rightarrow 4^2=(2^2)^2$

$16=2^4\rightarrow 16^3=(2^4)^3$

$32=2^5\rightarrow 32^2=(2^5)^2$

We can rewrite the original equation in these terms.

Simplify exponents.

$2^n=(2^4)(2^{12})(2^{10})$

Finally, combine terms.

$2^n=2^{4+12+10}=2^{26}$

From this equation, we can see that .

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Question

Which of the following is eqivalent to 5_b_ – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) , where b is a constant?

Answer

We want to simplify 5_b_ – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) .

Notice that we can collect the –5(b–1) terms, because they are like terms. There are 5 of them, so that means we can write –5(b–1) – 5(b–1) – 5(b–1) – 5(b–1) – 5(b–1) as (–5(b–1))5.

To summarize thus far:

5_b_ – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) – 5(_b–1) = 5_b +(–5(_b–_1))5

It's important to interpret –5(b–1) as (–1)5(b–1) because the –1 is not raised to the (b – 1) power along with the five. This means we can rewrite the expression as follows:

5_b_ +(–5(b–1))5 = 5_b_ + (–1)(5(b–1))(5) = 5_b_ – (5(b–1))(5)

Notice that 5(b–1) and 5 both have a base of 5. This means we can apply the property of exponents which states that, in general, abac = a b+c. We can rewrite 5 as 51 and then apply this rule.

5_b_ – (5(_b–1))(5) = 5_b – (5(_b–1))(51) = 5_b – 5(_b–_1+1)

Now, we will simplify the exponent b – 1 + 1 and write it as simply b.

5_b_ – 5(b–1+1) = 5_b* – 5_b* = 0

The answer is 0.

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Question

If $\dpi{100} \small r$ and $\dpi{100} \small s$ are positive integers, and $\dpi{100} \small 25\left ( 5^{r} \right )=5^{s-2}$ , then what is $\dpi{100} \small s$ in terms of $\dpi{100} \small r$ ?

Answer

$\dpi{100} \small 25\left ( 5^{r} \right )$ is equal to $5^{2}\left ( 5^{r}\right )$ which is equal to $\dpi{100} \small \left ( 5^{r+2} \right )$ . If we compare this to the original equation we get $\dpi{100} \small r+2=s-2\rightarrow s=r+4$

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Question

Solve for $x$ :

$4^{4}+4^{4}+4^{4}+4^{4}=2^{x}$

Answer

Combining the powers, we get $1024=2^{x}$ .

From here we can use logarithms, or simply guess and check to get $x=10$ .

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Question

If $x^2=11$ , then what does $x^4$ equal?

Answer

$x^4=(x^2)(x^2)=11\cdot 11=121$

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Question

Simplify. All exponents must be positive.

$\left ( x^{-2}y^{3} \right )\left ( x^{5}y^{-4} \right )$

Answer

Step 1: $\left ( x^{-2}x^{5} \right )= x^{3}$

Step 2: $\left ( y^{3}y^{-4} \right )= y^{-1}= \frac{1}{y}$

Step 3: (Correct Answer): $\frac{x^{3}}{y}$

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Question

Simplify. All exponents must be positive.

$\frac{x^{-3}y^{5}}{x^{2}y^{-1}}$

Answer

Step 1: $\frac{y^{5}}{\left ( x^{3}x^{2} \right )\left \right )y^{-1}}$

Step 2: $\frac{\left ( y^{5}y^{1} \right )}{x^{3}x^{2}}$

Step 3: $\frac{y^{6}}{x^{5}}$

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Question

$\frac{\left ( -11 \right )^{-8}}{\left ( -11\right )^{12}}$

Answer must be with positive exponents only.

Answer

Step 1: $\frac{1}{\left ( -11 \right )^{12}\left ( -11 \right )^{8}}$

Step 2: The above is equal to $\frac{1}{\left ( -11 \right )^{20}}$

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Question

Evaluate:

$-\left ( -3 \right )^{0}-\left ( -3^{0} \right )$

Answer

$-\left ( -3 \right )^{0}= -1$

$(-3^{0})=-1$

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Question

Simplify:

$3\sqrt{18} + 2\sqrt{50}$

Answer

$3\sqrt{18} = 3\sqrt{2\star 3^{2}} = 9\sqrt{2}$

Similarly $2\sqrt{50} = 2\sqrt{2\star 5^{2}} = 10\sqrt{2}$

So $9\sqrt{2} + 10\sqrt{2}= 19\sqrt{2}$

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Question

If $\frac{3^{y - 1}}{3^{-2}} = 27^{y}3^{y}$ , what is the value of ?

Answer

Rewrite the term on the left as a product. Remember that negative exponents shift their position in a fraction (denominator to numerator).

$3^{y-1}3^2=27^y3^y$

The term on the right can be rewritten, as 27 is equal to 3 to the third power.

$3^{y-1}3^2=(3^3)^y3^y$

Exponent rules dictate that multiplying terms allows us to add their exponents, while one term raised to another allows us to multiply exponents.

$3^{(y-1)+2}=(3)^{3y}3^y$

$3^{y+1}=3^{3y+y}=3^{4y}$

We now know that the exponents must be equal, and can solve for .

$\frac{1}{3}=y$

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Question

Simplify:

$x^{4}\cdot x^{5}$

Answer

When multiplying exponents with the same base, we use the rules of exponents.

This means you must simply add the exponents together as shown below:

$x^{4}\cdot x^{5} = x^{4+5} = x^{9}$

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Question

Simplify: $3^2\times3^3\times3^{-4}$

Answer

To determine the value of this expression, it is not necessary to determine the values of each term's power. Instead, since these powers have the same bases and are multiplied, the powers can be added.

$3^2\times3^3\times3^{-4} = 3^{2+3+(-4)} = 3^1=3$

The answer is .

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