# 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.

- 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,

- 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 | |

exponent is 1 | the result is the base | |

base is 0 | the result is 0 * | |

exponent is 0 | the result is 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.

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

- 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.

**Example 1**

Find the value of:

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?

= What number SQUARED is 36?

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

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

**Example 2**

Find the value of:

In this example, the BASE is the fraction 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 times 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.

Therefore we have which is written in lowest terms, so that is our final answer.

**Example 3**

Find the value of:

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 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.

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

**Example 4**

Find the value of:

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

= What number SQUARED is 36?

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

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

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.

**Example 5**

Estimate the value of:

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

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