Math

Question Find the value of xx in the equation 163=2x416^{3} = 2^{x} \cdot 4.

Studdy Solution

STEP 1

Assumptions
1. We are given the equation 163=2x416^{3}=2^{x} \cdot 4.
2. We need to solve for the value of xx.

STEP 2

First, we need to express all terms in the equation in terms of the same base to make the equation easier to solve. The base we will use is 2 because both 16 and 4 can be expressed as powers of 2.
16=2416 = 2^{4} 4=224 = 2^{2}

STEP 3

Now, rewrite the equation using these expressions.
(24)3=2x22\left(2^{4}\right)^{3} = 2^{x} \cdot 2^{2}

STEP 4

Apply the power of a power rule to simplify the left side of the equation. According to this rule, (am)n=amn(a^{m})^{n} = a^{m \cdot n}.
243=2x222^{4 \cdot 3} = 2^{x} \cdot 2^{2}

STEP 5

Simplify the exponent on the left side of the equation.
212=2x222^{12} = 2^{x} \cdot 2^{2}

STEP 6

On the right side of the equation, apply the product of powers rule, which states that aman=am+na^{m} \cdot a^{n} = a^{m+n}.
212=2x+22^{12} = 2^{x+2}

STEP 7

Now that we have the same base on both sides of the equation, we can equate the exponents. This is because if am=ana^{m} = a^{n}, then m=nm = n.
12=x+212 = x + 2

STEP 8

Solve for xx by isolating it on one side of the equation.
x=122x = 12 - 2

STEP 9

Calculate the value of xx.
x=10x = 10
The value of xx in the equation 163=2x416^{3}=2^{x} \cdot 4 is 10.

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