Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = 2 \)
Solution
To solve the equation:
\[
2^{x+2} + 2^{x} = 20
\]
**Step 1: Simplify the Equation**
First, recognize that \(2^{x+2}\) can be rewritten using the property of exponents \(a^{b+c} = a^{b} \cdot a^{c}\):
\[
2^{x+2} = 2^{x} \cdot 2^{2} = 4 \cdot 2^{x}
\]
Now, substitute this back into the original equation:
\[
4 \cdot 2^{x} + 2^{x} = 20
\]
**Step 2: Factor Out \(2^{x}\)**
Factor \(2^{x}\) from the left side of the equation:
\[
2^{x} (4 + 1) = 20
\]
\[
2^{x} \cdot 5 = 20
\]
**Step 3: Solve for \(2^{x}\)**
Divide both sides by 5 to isolate \(2^{x}\):
\[
2^{x} = \frac{20}{5}
\]
\[
2^{x} = 4
\]
**Step 4: Solve for \(x\)**
Recognize that 4 is a power of 2:
\[
4 = 2^{2}
\]
Therefore:
\[
2^{x} = 2^{2} \implies x = 2
\]
**Answer:** \( x = 2 \)
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Simplify this solution Bonus Knowledge
To solve the equation \( 2^{x+2} + 2^{x} = 20 \), we can rewrite \( 2^{x+2} \) as \( 4 \cdot 2^{x} \) (since \( 2^{x+2} = 2^2 \cdot 2^{x} = 4 \cdot 2^{x} \)). This gives us: \[ 4 \cdot 2^{x} + 2^{x} = 20 \] Now, factor out \( 2^{x} \): \[ (4 + 1) \cdot 2^{x} = 20 \] This simplifies to: \[ 5 \cdot 2^{x} = 20 \] Now, divide both sides by 5: \[ 2^{x} = 4 \] Since \( 4 = 2^{2} \), we can equate the powers of 2: \[ x = 2 \] Thus, the solution to the equation is \( x = 2 \).
