EXPLORING PROPERTIES OF INTEGER EXPONENTS

Properties of integer exponents can be used to solve many real-world problems. 

So, it is always important to explore the properties of integer exponents. 

A. Look at the following equations and answer the questions given below. 

3 · 3 · 3 · 3 · 3 = 3⁵

(3 · 3 · 3 · 3) · 3 = 3⁴ · 3¹ = 3⁵

(3 · 3 · 3) · (3 · 3) = 3³ · 3² = 3⁵

1. What pattern do you see when multiplying two powers with the same base ?

The result has the same base with an exponent equal to the sum of the exponents in the powers.

2. Use your pattern to find the value of 'n' in the equation given below.

5²· 5³ = 5ⁿ -----(1)

We know the fact that the result has the same base with an exponent equal to the sum of the exponents in the powers. 

So, we have 

5²· 5³ = 5⁵ -----(2)

Comparing (1) and (2), we get n  =  5.

B. Look at the following equation and answer the question given below.  

1. What pattern do you see when dividing two powers with the same base ?

The result has the same base with an exponent equal to the difference of the exponent in the numerator and exponent in the denominator.

2. Use your pattern to find the value of n in the equation given below.

6⁸/6³ = 6ⁿ -----(1)

We know the fact that the result has the same base with an exponent equal to the difference of the exponent in the numerator and exponent in the denominator.

So, we have 

6⁸/6³ = 6⁵-----(2)

Comparing (1) and (2), we get n  =  5.

C. Look at the following equation and answer the questions given below. 

(5³)² = 5³ · 5³

(5³)² = 5³⁺³

(5³)² = 5⁶

1. What pattern do you see when raising a power to a power ?

The result has the same base with an exponent equal to the product of the exponents.

2. Use your pattern to find the value of n in the equation given below.

(7²)⁵ = 7ⁿ -----(1)

We know the fact that the result has the same base with an exponent equal to the product of the exponents.

So, we have 

(7²)⁵ = 7¹⁰ -----(2)

Comparing (1) and (2), we get n = 10.

Reflect

1. General rule for the value of a^m ⋅ aⁿ.

a^m ⋅ aⁿ = a^{m + n}

2. General rule for the value of a^m ÷ aⁿ. a ≠ 0.

a^m ÷ aⁿ = a^{m - n}

3. General rule for the value of (a^m)ⁿ.

(a^m)ⁿ = a^mn

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