QuestionProve that $2^{4x+1} + 32^{\frac{4}{5}(x+1)}$ is divisible by 9.

Studdy Solution

STEP 1

Assumptions1. We need to show that the expression $^{4 x+1}+32^{\frac{4}{5}(x+1)}$ is divisible by9. We can use the properties of exponents and the divisibility rule for9 to solve this problem

STEP 2

First, we need to simplify the expression. We can do this by expressing32 as a power of2.
$32 =2^5$

STEP 3

Substitute $2^5$ for32 in the expression.
$2^{ x+1}+32^{\frac{}{5}(x+1)} =2^{ x+1}+(2^5)^{\frac{}{5}(x+1)}$

STEP 4

implify the expression using the property of exponents $a^{mn} = (a^m)^n$ .
2^{4 x+1}+(2^)^{\frac{4}{}(x+1)} =2^{4 x+1}+2^{4(x+1)}

STEP 5

Now, we have two terms in the form $2^n$ . We can rewrite these terms using the property of exponents $a^{m+n} = a^m \times a^n$ .
$2^{4 x+1}+2^{4(x+1)} =2^{4x} \times2^1 +2^{4x} \times2^4$

STEP 6

implify the expression.
$2^{4x} \times2^1 +2^{4x} \times2^4 =2^{4x+1} +2^{4x+4}$

STEP 7

Now, we can factor out $2^{4x}$ from both terms.
$2^{4x+1} +2^{4x+4} =2^{4x}(2 +2^4)$

STEP 8

implify the expression.
$2^{4x}(2 +2^4) =2^{4x} \times18$

STEP 9

Now, we can see that the expression is a multiple of9, because18 is a multiple of9. Therefore, the expression is divisible by9.

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