Certainly! Let's start by analyzing the given expression [tex]\( 4^7 \div 4^9 \)[/tex].
### Step 1: Simplify the Expression Using Properties of Exponents
We can use the property of exponents that states [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex]. Here, [tex]\( a = 4 \)[/tex], [tex]\( m = 7 \)[/tex], and [tex]\( n = 9 \)[/tex].
So,
[tex]\[ 4^7 \div 4^9 = \frac{4^7}{4^9} \][/tex]
Applying the exponent rule:
[tex]\[ \frac{4^7}{4^9} = 4^{7-9} \][/tex]
Simplify the exponent:
[tex]\[ 7 - 9 = -2 \][/tex]
Thus,
[tex]\[ \frac{4^7}{4^9} = 4^{-2} \][/tex]
### Step 2: Interpret the Negative Exponent
By definition, a negative exponent represents the reciprocal of the base raised to the positive version of the exponent. In this case:
[tex]\[ 4^{-2} = \frac{1}{4^2} \][/tex]
Let's break this down to show the steps explicitly:
1. The negative exponent [tex]\(-2\)[/tex] indicates that we need to take the reciprocal of [tex]\(4^2\)[/tex].
2. Calculate [tex]\(4^2\)[/tex]:
[tex]\[ 4^2 = 4 \times 4 = 16 \][/tex]
3. Now, take the reciprocal:
[tex]\[ \frac{1}{4^2} = \frac{1}{16} \][/tex]
### Step 3: Final Verification
Now, let's verify that our interpretation holds true for the values:
First, calculate [tex]\(4^{-2} \)[/tex]:
[tex]\[ 4^{-2} = \frac{1}{4^2} \][/tex]
We already know that:
[tex]\[ 4^2 = 16 \][/tex]
So,
[tex]\[ \frac{1}{4^2} = \frac{1}{16} \][/tex]
Therefore, the value of [tex]\( 4^{-2} \)[/tex] is indeed:
[tex]\[ 4^{-2} = \frac{1}{16} \][/tex]
### Conclusion
Hence, we have shown that:
[tex]\[ 4^7 \div 4^9 = 4^{-2} \][/tex]
and that
[tex]\[ 4^{-2} = \frac{1}{4^2} \][/tex]
Everything checks out mathematically. This illustrates the property of negative exponents and their relationship to reciprocals effectively.
