GED Mathematical Reasoning: Powers And Roots I

Exponent

  • Exponents are a shorthand way of writing repeated multiplication.
  • The value in the ‘2’ spot is called the BASE. The number written in the ‘5’ spot is what we call the EXPONENT or POWER.

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  • The base tells us what to multiply while the exponent tells us how many times to write the base in the product. In other words, 2^5 means: 2 times 2 times 2 times 2 times 2

2^5 = 2 \times 2 \times 2 \times 2 \times 2

  • When faced with a negative exponent, rewrite the expression as a fraction over one. Then, to find the final answer: flip the fraction, make the exponent positive, and simplify.
  • We have special wording when the exponent is 2 or 3. When a base is raised to a power of 2, we say it is “squared”. When a base is raised to a power of 3, we say it is “cubed”.
  • Before we move on to discuss “roots”, let’s talk briefly about bases and exponents involving one or zero. You may follow along with this chart while I read:
If the… Then… EXAMPLE
base is 1 the result is 1 1^3 = 1 \times 1 \times 1 = 1
exponent is 1 the result is the base 4^1 = 4
base is 0 the result is 0 * 0^4 = 0 \times 0 \times 0 \times 0 = 0
exponent is 0 the result is 1 * 7^0 = 1
  • If the base is 1, the result will be 1 – no matter what the exponent.
  • If the exponent is 1, then the result will be equal to the base. This is related to an important side note: if an exponent is not shown, assume the exponent is ONE.
  • If the base is zero, then the result will be zero – no matter what the exponent.
  • And if the exponent is zero, then the result will be ONE – no matter what the base.
  • The only exception to this is if zero is raised to a power of zero. In that case, we say the answer is “indeterminate” – meaning that there is no one right answer. If you try to use the calculator to find this value, you will receive an “error” message.

 

Square root

  • Just like addition and subtraction are opposite operations, to square root a number is the opposite of squaring the number.
  • So to find the “square root” of 36, we’ll ask ourselves: What number SQUARED is 36?
  • Down the road, you will learn about higher order roots, but for now we will focus on the SQUARE root. Technically, the square root sign has a little number “2” sitting in what we call the “root position” However, when the root is a 2, we typically don’t write it.

\sqrt{36} = \sqrt[2]{36}

  • It is handy to have a table of what we call “perfect squares,” which are numbers whose square root is a whole number.

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  • Using this reference, it is easy to see – for example – that the square root of 144 is 12, since 12 squared is equal to 144.

\sqrt{144} = 12, since 12^2 = 144

 

Example 1

Find the value of: 2^5

 

To square root a number is the opposite of squaring the number. So to find the “square root” of 36, we’ll ask ourselves: What number SQUARED is 36?

\sqrt{36} = What number SQUARED is 36?

The answer is 6. So we say that the square root of 36 equals 6.

\sqrt{36} = 6

To find the value of 2^5, we’ll multiply the base by the exponent. The result is 32.

2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32

 

Example 2

Find the value of: {\frac{3}{4}}^2

 

In this example, the BASE is the fraction \frac{3}{4} and the EXPONENT is 2, which is why I read it as “three-fourths squared”. Sometimes we use parenthesis around the base if it contains a negative number or a fraction.

We’ll write \frac{3}{4} times \frac{3}{4} and then multiply the fractions together by multiplying straight across. 3 times 3 is 9, which we’ll place in the numerator of our answer. 4 times 4 is 16, which we’ll place in the denominator of our answer.

{\frac{3}{4}}^2 = \frac{3}{4} \times \frac{3}{4} = \frac{3 \times 3}{4 \times 4} = \frac{9}{16}

Therefore we have  \frac{9}{16} which is written in lowest terms, so that is our final answer.

 

Example 3

Find the value of: 5^{-3}

 

When faced with a negative exponent, rewrite the expression as a fraction over one. Then, flip the fraction, make the exponent positive, and simplify.

So first, we’ll rewrite 5^{-3} as a fraction over one. When we flip the fraction and make the exponent positive, we have one over five to the power of positive three.

5^{-3} \rightarrow \frac{5^{-3}}{1} \rightarrow \frac{1}{5^3}

From there, we can simplify the denominator, five cubed. Since 5 times 5 times 5 is 125, the final answer is ‘one over 125’.

5^{-3} = \frac{1}{5^3} = \frac{1}{5 \cdot 5 \cdot 5} = \frac{1}{125}

 

Example 4

Find the value of: \sqrt{36}

 

To square root a number is the opposite of squaring the number. So what number SQUARED is 36?

\sqrt{36} = What number SQUARED is 36?

The answer is 6. So we say that the square root of 36 equals 6.

\sqrt{36} = 6

It is handy to have a table of what we call “perfect squares,” which are numbers whose square root is a whole number.

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Using this reference, it is easy to see – for example – that the square root of 144 is 12, since 12 squared is equal to 144.

\sqrt{144} = 12, since 12^2 = 144

 

Example 5

Estimate the value of: \sqrt{56}

 

56 is not a perfect square but it’s about halfway between 49 and 64, which ARE perfect squares. The square root of 49 is 7 and the square root of 64 is 8. So since 56 is about halfway between 49 and 64, then its square root is about halfway between 7 and 8. So we can say that the square root of 56 is APPROXIMATELY equal 7.5

\sqrt{56} \approx 7.5

Notice that we use a squiggly equal sign to denote that the square root of 56 is approximately equal to 7.5

 

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