How to add exponents - GRE Quantitative Reasoning

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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

Indicate whether Quantity A or Quantity B is greater, or if they are equal, or if there is not enough information given to determine the relationship.

$\dpi{100} \small n>0$

Quantity A: $\dpi{100} \small 16^{n+2}$

Quantity B: $\dpi{100} \small 2^{4}\times (8^{n+1})^{2}\div 4^{n}$

Answer

By using exponent rules, we can simplify Quantity B.

$\dpi{100} \small \dpi{100} \small 2^{4}\times (8^{n+1})^{2}\div 4^{n}$

$\dpi{100} \small \dpi{100} \small 2^{4}\times (8^{2n+2})\div 4^{n}$

$\dpi{100} \small \dpi{100} \small 2^{4}\times 2^{3(2n+2)}\div 4^{n}$

$\dpi{100} \small \dpi{100} \small 2^{4}\times 2^{6n+6}\div 4^{n}$

$\dpi{100} \small \dpi{100} \small 2^{6n+10}\div 4^{n}$

$\dpi{100} \small \dpi{100} \small 2^{6n+10}\div 2^{2n}$

$\dpi{100} \small 2^{4n+10}$

Also, we can simplify Quantity A.

$\dpi{100} \small 16^{n+2}$

$\dpi{100} \small =2^{4(n+2)}$

$\dpi{100} \small =2^{4n+8}$

Since n is positive, $\dpi{100} \small 4n+10>4n+8$

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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 $x^{^{3}}x^{^{5}} - x^{^{2}} + (y^{^{2}})^{^{2}}$ .

Answer

First, simplify $x^{3}x^{5}$ by adding the exponents to get $x^{8}$ .

Then simplify $(y^{2})^{2}$ by multiplying the exponents to get $y^{4}$ .

This gives us $x^{8} - x^{^{2}} + y^{4}$ . We cannot simplify any further.

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Question

Simplify $x^{3}x^{6} + (x^{3})^{3} + x^{3}x^{2}x^{2}x^{2}$ .

Answer

Start by simplifying each individual term between the plus signs. We can add the exponents in and so each of those terms becomes . Then multiply the exponents in so that term also becomes . Thus, we have simplified the expression to which is $3x^{9}$ .

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Question

If $3^4 \cdot 3^8 = 9^x$ , what is the value of

Answer

To attempt this problem, note that .

Now note that when multiplying numbers, if the base is the same, we may add the exponents:

$3^4 \cdot 3^8 = 3^{4+8}=3^{12}$

This can in turn be written in terms of nine as follows (recall above)

$3^{12}=9^{\frac{12}{2}}=9^6$

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Question

If $(3^2\cdot3^5)^3=3^x$ , what is the value of

Answer

When dealing with exponenents, when multiplying two like bases together, add their exponents:

$3^2\cdot3^5=3^7$

However, when an exponent appears outside of a parenthesis, or if the entire number itself is being raised by a power, multiply:

$(3^7)^3=3^{7\cdot 3}=3^{21}$

$3^{21}=3^{x}$

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