Exponent Practice Problems – Test 2

  1. Which of the following is equal to \(\dfrac{4^6}{2^8}\) ?
  1. 2
  2. 8
  3. 16
  4. 32
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The correct answer is C!

Since 4 is the same as 22, \(4^6 = (2^2)^6 = 2^{12}\).

When dividing exponential numbers with the same base, simply subtract the exponent in the denominator from the exponent in the numerator:

\(\dfrac{2^{12}}{2^8}=2^{12-8}=2^4=16\)

 

  1. Which of the following is equal to \(x^{mn}\) ?
  1. \((x^m)^n\)
  2. \(x^{m+n}\)
  3. \(x^mx^n\)
  4. \(nx^m\)
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The correct answer is A!

To determine the power of a power, multiply the exponents. This example presents the reverse case: the product of exponents is equivalent to the power of a power.

 

  1. There are 64 squares on a checkerboard. Bobby puts one penny on the first square, two on the second square, four on the third, eight on the fourth, and continues to double the number of coins at each square until he has covered all 64 squares. How many coins must he place upon the last square?
  1. \(2^{64}\)
  2. \(2^{64}-1\)
  3. \(2^{63}\)
  4. \(2^{63}+1\)
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The correct answer is C!

The number of coins on each square represents consecutive powers of 2, since the number doubles with each consecutive square. However, the series of powers begins with 0 for the first square, so that for the 64th square, the number of coins will be 263.

 

  1. Simplify the following expression:
\(\dfrac{50x^{18}t^6w^3z^{20}}{5x^5t^2w^2z^{19}}\)
  1. \(10x^{13}t^3wz\)
  2. \(10x^{13}t^4wz\)
  3. \(10x^{12}t^4wz\)
  4. \(10x^{13}t^4wz^2\)
Show Answer
The correct answer is B!

To simplify this expression, it is necessary to follow the law of exponents that states \(\tfrac{x^n}{x^m}\) \(=x^{n-m}\).

First, the 50 can be divided by 5:

\(50 \div 5 = 10\)

Then, it’s simply a matter of using the law of exponents described above to simplify the expression:

\((50x^{18}t^6w^3z^{20}) \div (5x^5t^2w^2z^{19})\)

\(= 10x^{18-5}t^{6-2}w^{3-2}z^{20-19}\)

\(= 10x^{13}t^4wz\)

 

  1. Simplify the following expression:
\((3x^2\times 7x^7)+(2y^3\times 9y^{12})\)
  1. \(21x^{14}+18y^{26}\)
  2. \(10x^9+11y^{15}\)
  3. \(21x^{14}+18y^{15}\)
  4. \(21x^9+18y^{15}\)
Show Answer
The correct answer is D!

To simplify this expression, it is necessary to follow the law of exponents that states \(x^nx^m =x^{n+m}\).

Therefore:

\((3x^2\times 7x^7)+(2y^3\times 9y^{12})\)

\(= 21x^{7+2}+18y^{12+3}\)

\(= 21x^9+18y^{15}\)

 

  1. Simplify the following expression:
\((2x^4y^7m^2z)\times (5x^29^3m^8)\)
  1. \(10x^6y^9m^{10}z\)
  2. \(7x^6y^{10}m^{10}z\)
  3. \(10x^5y^{10}m^{10}z\)
  4. \(10x^6y^{10}m^{10}z\)
Show Answer
The correct answer is D!

To simplify this expression, it is necessary to follow the law of exponents that states \(x^mx^n =x^{m+n}\).

Therefore:

\((2x^4y^7m^2z) \times (5x^2y^3m^8)\)

\(= 10x^{4+2}y^{7+3}m^{2+8}z\)

\(= 10x^6y^{10}m^{10}z\)

 

  1. If \(2^4=4^x\), then \(x =\) ?
  1. 2
  2. 4
  3. 6
  4. 8
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The correct answer is A!

We can write \(2^4\) as:

\(2 \times 2 \times 2 \times 2=16\)

Therefore, since \(4^x=16\), we know that \(x=2\).

 

  1. If \(3^4=9^x\), then \(x =\) ?
  1. 2
  2. 4
  3. 6
  4. 8
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The correct answer is A!

We can write \(3^4\) as:

\(3 \times 3 \times 3 \times 3=81\)

Therefore, since \(9^x=81\), we know that \(x=2\).

 

  1. If \(\tfrac{6^x}{6^2+6^2+6^2}\) \(=\) \(\tfrac{1}{3}\), then \(x =\) ?
  1. 2
  2. 4
  3. 6
  4. 8
Show Answer
The correct answer is A!

The denominator is equal to \(3 \times 6^2\), so that the expression becomes:

\(\dfrac{6^x}{6^2+6^2+6^2}=\dfrac{6^x}{3\times 6^2}=\dfrac{1}{3}\)

If \(x = 2\) so that \(6^x=6^2\), these will cancel on top and bottom, leaving \(\tfrac{1}{3}\).