Question
Use the like bases property to solve the equation \( 64^{x}=1024 \) \( x=\square \)
Ask by Mills Riley. in the United States
Mar 13,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = \frac{5}{3} \)
Solution
To solve the equation \( 64^{x} = 1024 \) using the like bases property, we first express both sides of the equation with the same base.
1. **Identify the bases**:
- \( 64 \) can be expressed as \( 4^3 \) because \( 4^3 = 64 \).
- \( 1024 \) can be expressed as \( 4^5 \) because \( 4^5 = 1024 \).
2. **Rewrite the equation**:
\[
(4^3)^{x} = 4^5
\]
3. **Apply the power of a power property**:
\[
4^{3x} = 4^5
\]
4. **Set the exponents equal to each other** (since the bases are the same):
\[
3x = 5
\]
5. **Solve for \( x \)**:
\[
x = \frac{5}{3}
\]
Thus, the solution is:
\[
x = \frac{5}{3}
\]
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Simplify this solution Bonus Knowledge
To solve the equation \( 64^{x} = 1024 \), we start by expressing both numbers as powers of the same base. Notably, both 64 and 1024 can be expressed as powers of 2: - \( 64 = 2^6 \) - \( 1024 = 2^{10} \) Now we can rewrite the equation using these bases: \((2^6)^{x} = 2^{10}\) Using the property of exponents \((a^{m})^{n} = a^{mn}\), we get: \(2^{6x} = 2^{10}\) Since the bases are the same, we can set the exponents equal to each other: \(6x = 10\) Now, we solve for \(x\): \(x = \frac{10}{6} = \frac{5}{3}\) So, the solution is: \(x = \frac{5}{3}\)
