Properties of Exponents - SSAT Upper Level Quantitative

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Question

Evaluate: $\dpi{100} (3^{3})^{2}$

Answer

A power raised to a power indicates that you multiply the two powers.

$\dpi{100} (3^{3})^{2}=3^{3\cdot 2}=3^{6}$

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Question

Evaluate $\dpi{100} \frac{2^{10}}{2^{8}}$

Answer

If you divide two exponential expressions with the same base, you can simply subtract the exponents. Here, both the top and the bottom have a base of 2 raised to a power.

So $\dpi{100} \frac{2^{10}}{2^{8}}=2^{10-8}=2^{2}=4$

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Question

$\dpi{100} 2^{3}\cdot 2^{2}$

Answer

Since the two expressions have the same base, we just add the exponents.

$\dpi{100} 2^{3}\cdot 2^{2}=2^{3+2}=2^{5}=32$

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Question

$\dpi{100} Evaluate: (0.50^{2})$

Answer

We can either write $\dpi{100} 0.5\times 0.5$ , or we can convert this to a fraction and write

$\dpi{100} \frac{1}{2}\times \frac{1}{2}$

$\dpi{100} \frac{1}{2}\times \frac{1}{2}=\frac{1\times 1}{2\times 2}=\frac{1}{4}$

$\dpi{100} \frac{1}{4}$ in decimal form is 0.25.

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Question

If $\dpi{100} x=\sqrt{34}$ then

Answer

$\dpi{100} \sqrt{25}=5$

$\dpi{100} \sqrt{36}=6$

34 is between 25 and 36, thus its square root must be between the square roots of 25 and 36. x must be a number between 5 and 6.

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Question

$\dpi{100} (0.75)^{2}=$

Answer

Convert .75 to a fraction. $\dpi{100} \frac{75}{100}=\frac{3}{4}$ .

Now multiply $\dpi{100} \frac{3}{4}\times \frac{3}{4}=\frac{3\times 3}{4\times 4}=\frac{9}{16}$

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Question

and are both positive integers; A is odd. What can you say about the number

$(A + B + 1)^{A+B+1}$ ?

Answer

If is odd, then , the sum of three odd integers, is odd; an odd number taken to any positive integer power is odd.

If is even, then , the sum of two odd integers and an even integer, is even; an even number taken to any positive integer power is even.

Therefore, $(A + B + 1)^{A+B+1}$ always assumes the same odd/even parity as .

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Question

Express 0.00000000000097 in scientific notation.

Answer

To rewrite a very small number in scientific notation:

Write the number.

Move the decimal point right as many places as needed until it follows the first nonzero digit, which here is the nine. Count the number of places it is moved - here it will be thirteen places.

The number formed is , which will be placed in front; , the negative of the number of places counted, will be the exponent. The number, in scientific notation, will be $9.7 \times 10 ^{-13}$ .

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Question

$(-4)^{2} + (-4)^{3}$

Answer

$(-4)^{2} = -4 \cdot (-4) = +(4\cdot 4) = 16$

$(-4)^{3} = -4 \cdot (-4)\cdot (-4) = 16 \cdot (-4) = - (16 \cdot 4) = -64$

Therefore,

$(-4)^{2} + (-4)^{3} = 16 + (-64) = - (64-16) = -48$

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Question

Simplify:

$\frac{8x ^{-2}}{(3x)^{2}}$

Answer

$\frac{8x ^{-2}}{(3x)^{2}}$

To solve this problem, we start with the parentheses and exponents in the denominator.

$= \frac{8x ^{-2}}{3^{2}x^{2}} = \frac{8x ^{-2}}{9x^{2}}$

Next, we can bring the from the denominator up to the numerator by making the exponent negative.

$= \frac{8x ^{-2-2}}{9} = \frac{8x ^{-4}}{9}$

Finally, to get rid of the negative exponent we can bring it back down to the denominator.

$= \frac{8}{9x ^{4}}$

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Question

Evaluate:

$\frac{\left ( -5\right ) ^{5} - 5^{5}}{\left ( -5\right ) ^{5} + 5^{5}}$

Answer

$5^{5} = 5 \times 5 \times 5 \times 5 \times 5 = 3,125$

$\left (-5 \right ) ^{5} = -\left ( 5^{5} \right ) = -3,125$

$\frac{\left ( -5\right ) ^{5} - 5^{5}}{\left ( -5\right ) ^{5} + 5^{5}} = \frac{-3,125 -3,125}{-3,125 + 3,125} =\frac{-6,250}{0}$

This is an undefined quantity.

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Question

Evaluate:

$\frac{4^{4}}{4^{4} +\left ( -4 \right ) ^{4} }$

Answer

$4^{4} = 4 \times 4 \times 4 \times 4 = 256$

$\left (- 4 \right ) ^{4} = \left (- 4 \right ) \times \left (- 4 \right ) \times \left (- 4 \right ) \times \left (- 4 \right ) = 256$

$\frac{4^{4}}{4^{4} +\left ( -4 \right ) ^{4} } = \frac{256 }{256 + 256} = \frac{256 }{512} =\frac{1 }{2}$

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Question

Evaluate:

$(100 - 4 \cdot 25) ^{100 \div 25 }$

Answer

$(100 - 4 \cdot 25) ^{100 \div 25 } = (100 - 100) ^{4} = 0 ^{4} =0$

as 0 taken to any positive power is equal to 0.

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Question

Evaluate:

$(80 - 4 \cdot 25) ^{100 \div 25 }$

Answer

$(80 - 4 \cdot 25) ^{100 \div 25 }$

$= (80 - 100) ^{4 }$

$= (-20) ^{4 }$

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Question

Assume all variables to be nonzero.

Simplify: $6 (xyz)^{0} + 4(xyz)^{0}$

Answer

Any nonzero expression to the power of zero is equal to 1:

$6 (xyz)^{0} + 4(xyz)^{0} = 6 \times 1 + 4 \times 1 = 6 + 4 = 10$

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Question

Express the result in scientific notation:

$\left ( 3 \times 10 ^{4}\right )^{3}$

Answer

Applying the power of a product property of exponents, then the power of a power property:

$\left ( 3 \times 10 ^{4}\right )^{3} = 3^{3}\times\left ( 10 ^{4}\right )^{3} = 27 \times 10 ^{4 \times3} = 27 \times 10 ^{12}$

This is not in scientific notation, so we adjust it as follows, applying the product of powers property:

$27 \times 10 ^{12} = 2.7 \times 10 ^{1} \times 10 ^{12} = 2.7 \times 10 ^{1+ 12} =2.7 \times 10 ^{13}$

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Question

Assume all variables to be nonzero.

Simplify: $4 \left ( xyz \right ) ^{3} (xyz) ^{0}$

Answer

Since any nonzero expression taken to the power of 0 is equal to 1,

$4 \left ( xyz \right ) ^{3} (xyz) ^{0} = 4 \left ( xyz \right ) ^{3} \cdot 1 = 4 \left ( xyz \right ) ^{3}$

Using the power of a product property of exponents, this becomes

$4 \left ( xyz \right ) ^{3} = 4 x^{3}y^{3}z^{3}$

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Question

Express the result in scientific notation:

$(2 \times 10^{3}) (3 \times 10^{4}) (5 \times 10^{5})$

Answer

Apply the product of powers property:

$(2 \times 10^{3}) (3 \times 10^{4}) (5 \times 10^{5})$

$= 2 \times 3 \times 5 \times 10^{3} \times 10^{4} \times 10^{5}$

$= 30 \times 10^{3+4+5}$

$= 30 \times 10^{12}$

Since this is not in scientific notation, adjust as follows:

$30 \times 10^{12}$

$= 3 \times 10^{1} \times 10^{12}$

$= 3 \times 10^{1+12}$

$= 3 \times 10^{13}$

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Question

Express the result in scientific notation:

$\frac{1}{2.5 \times 10 ^{5}}$

Answer

Rewriting the numerator and applying the quotient of powers property:

$\frac{1}{2.5 \times 10 ^{5}}$

$= \frac{1\times 10 ^{0}}{2.5 \times 10 ^{5}}$

$= \frac{1 }{2.5 } \times \frac{ 10 ^{0}}{ 10 ^{5}}$

$= 0.4 \times 10 ^{0-5}$

$= 0.4 \times 10 ^{ -5}$

Since this is not in scientific notation, adjust as follows:

$0.4 \times 10 ^{ -5}$

$= 4\times 10 ^{ -1} \times 10 ^{ -5}$

$= 4\times 10 ^{ -1 + (-5)}$

$= 4\times 10 ^{ -6}$

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Question

Express the result in scientific notation:

$\frac{1}{1.25 \times 10^{-6}}$

Answer

Rewriting the numeator and applying the quotient of powers property:

$\frac{1}{1.25 \times 10^{-6}}$

$= \frac{1 \times 10^{0}}{1.25 \times 10^{-6}}$

$= \frac{1 }{1.25 } \times \frac{ 10^{0}}{ 10^{-6}}$

$=0.8 \times 10^{0- (-6)}$

$=0.8 \times 10^{6}$

This is not in scientific notation, so we adjust it as follows, applying the product of powers property:

$0.8 \times 10^{6}$

$=8 \times 10^{-1} \times 10^{6}$

$=8 \times 10^{-1+6}$

$=8 \times 10^{5}$

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