Lesson 8 Instruction, Page 1

Pre-Test		Post-Test
	Using Positive Exponents

The Rules of Exponents
CA GR7 NS 2.1, CA GR7 AF 2.1

As we have noted, exponents are used to simplify math terms. For example, we can write 5³ instead of 5 x 5 x 5. Both of the above terms have the same value, which is 125. And when working math problems with terms that have exponents, we normally leave the terms in the form with the exponents, instead of multiplying the terms out. We do this because it is much simpler and saves us from doing several calculations. In doing so, however, there are certain rules with exponents that must be followed so that all values will remain accurate, and allow us to keep all of them in the simplest forms.

In the following situations, the rules bellow will apply:

1. The product of LIKE factors with EXPONENTS.

5² • 5³ = ?

RULE: WHEN YOU MULTIPLY LIKE NON-ZERO BASE NUMBERS (FACTORS) THAT HAVE EXPONENTS: ADD THE EXPONENTS, and don't change the base number.

So 5² • 5³ becomes 5⁵

because

5² • 5³ means 5 x 5 x 5 x 5 x 5.

Now try these:

6³ · 6⁴ =

7² • 7⁴ • 7³ =

9² • 9⁴ • 9⁵ • 9⁶ =

and the answers are:

6⁷, 7⁹, and 9¹⁷.

So remember: When like (non-zero) base numbers with exponents are multiplied, the exponents are added, and the base numbers remain the same.

2. The quotient of LIKE dividends/divisors with EXPONENTS.

5⁴ • 5² = ?

RULE: WHEN YOU DIVIDE LIKE NON-ZERO BASE NUMBERS (DIVIDENDS BY DIVISORS) THAT HAVE EXPONENTS, SUBTRACT THE EXPONENT OF THE DIVISOR FROM THE EXPONENT OF THE DIVIDEND, and don't change the BASE NUMBER.

So becomes 5²

because

means

and whether you cancel the terms of the fraction, or divide 625 by 25, you get

25 or 5².

Now try these:

=

=

=

and the answers are:

6², 7¹ (or 7), and 8².

So remember: When like (non-zero) base numbers with exponents are divided, the divisor exponent is subtracted form the dividend exponent, and the base number remains the same (raised to the power of the difference).

3. A (non-zero) base number with an exponent raised to a power.

(5³)² = ?

RULE: MULTIPLY THE EXPONENTS.

So (5³)² becomes 5⁶

because:

(5³)² means (5³) (5³) or

5 x 5 x 5 x 5 x 5 x 5.

Now try these:

(3⁴)² =

(7³)³ =

(9²)² =

and the answers are:

3⁸, 7⁹, and 9⁴.

So remember: when you raise a (non-zero) base number with an exponent to a power, MULTIPLY THE EXPONENTS, and the base number remains the same.

4. A (non-zero) product with two or more factors is raised to a power.

(4 • 5)² = ?

RULE: THE EXPONENT BELONGS TO EACH OF THE FACTORS WHICH MEANS THAT EACH FACTOR SHOULD BE RAISED TO THE POWER.

So (4 · 5)² becomes 4² • 5²

which is

(4²) (5²) = 16 • 25, or 400.

5. A fraction (with non-zero digits) raised to a power.

= ?

RULE: THE EXPONENT BELONGS TO EACH OF THE TERMS (THE NUMERATOR AND THE DENOMINATOR).

So becomes

which is

9 ÷ 16 = 0.5625.

Now try these:

=

=

=

and the answers are:

, , and .

So remember: When a (non-zero) fraction is raised to a power, the exponent belongs to both the numerator and the denominator.

6. A term with a negative exponent.

5^-2 = ?

RULE: CHANGE THE TERM TO ITS RECIPROCAL AND CHANGE THE SIGN OF THE EXPONENT (TO A PLUS).

So 5^-2 becomes

because from Rule 2 (above), you know that

= 5^-2

which also equals

and = .

Now try converting these numbers with negative exponents:

6^-2 =

3^-4 =

4^-5 =

and the answers are:

(or ), (or ) and (or ).

So remember: Any term with a negative exponent can be changed to its reciprocal if you change the sign of the exponent to a plus.

7. The value of terms with EXPONENTS of ZERO.

7⁰ = ?

RULE: ANY TERM WITH AN EXPONENT OF ZERO HAS THE VALUE OF ONE.

So 7⁰ becomes 1.

Because: Whenever we divide a number by itself, the answer is 1 and since equals 1, and also equals 7⁰ (as shown in Rule 2 above).

So 7⁰ = 1

Now try these:

8⁰ =

16⁰ =d

1,456.675⁰ =

and the answers are all1.

So remember: Any number or term raised to the ZERO POWER is equal to 1.

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