Exponents: Exponential notation and roots

When we multiply a number by itself several times it can become cumbersome to write out the multiplication. For example, if we want to multiply 3 by itself 20 times, that is: $3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3$ (Imagine if you were multiplying it 100 times!)

Instead of writing out this long multiplication, we can use a short form called exponential notation. In this example, $3^{20}$ where 3 is called the base and 20 is called the exponent or power.

Therefore,

3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^{20}

Observe that the factor (or the number) that we are multiplying is called the 'base' and the number of times we are multiplying the factor is indicated by the exponent.

You can use your calculator to evaluate exponential terms.Now let us take a look at some examples:

Example 1: Write $5 \times 5 \times 5 \times 5 \times 5 \times 5$ in exponential notation.

Solution:

In this example, 5 is multiplied by itself six times, so the base is 5 and the exponent is 6. Hence this multiplication in exponential form will look like:

5^{6}

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Example 2: Evaluate $7^{4}$

Solution:

7^{4} = 7 \times 7 \times 7 \times 7 = 2401

This video demonstrates the calculator keys used to evaluate exponential terms.

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Example 3: Evaluate and compare $- 3^{6} a n d {(- 3)}^{6}$

Solution:

First of all let us examine

- 3^{6}

Here 3 has been multiplied times itself 6 times and the answer multiplied by -1, that is:

- 3^{6} = - 1 \times 3^{6}

= - 1 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3

= - 1 \times 729

= - 729

Now let us take a look at ${(- 3)}^{6}$

this term clearly has a base of -3 and an expontent of 6, so it can be evaluated in the following way:

${(- 3)}^{6} = (- 3) \times (- 3) \times (- 3) \times (- 3) \times (- 3) \times (- 3)$

= 729

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Roots

Roots, such as square roots and cubic roots, are another form of exponents. For instance, if I wish to determine what number, when multiplied times itself 2 times gives an answer of 16, then I am solving for the square root, or second root of 16.

Or $? \times ? = 16$

Let us assume that number is $a$ , so mathematically this situation can be represented as

$a = \sqrt[2]{16}$ or simply $a = \sqrt{16}$

We know that $4 \times 4 = 16$ therefore, $a = 4$

We can also use our calculator to find this solution. Here , 4 is called the square root of 16.

This video demonstrates the calculator keys used to evaluate square roots.

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Similarly, we can find the cubic root of a number and the nth root of a number.

Let us find the cubic root of 125. Symbolically this problem can be expressed as:

Example 4: Find $\sqrt[3]{125}$

Solution:

Again we know that

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

Therefore,

\sqrt[3]{125} = 5

Use the cubic root key of your calculator to evaluate this problem.

This video demonstrates the calculator keys used to evaluate cubic roots.

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Example 5: Find $\sqrt[5]{32}$

Solution:

In this question we are interested in finding the 5th root of 32 and we know that

2^{5} = 32

Hence,

\sqrt[5]{32} = 2

Use the nth root key of your calculator to evaluate this problem.

This video demonstrates the calculator keys used to evaluate the nth root of any given number.

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