To solve the equation [tex]\(\sqrt{5^2 + 12^2} = \sqrt[3]{64a^3}\)[/tex], we will work through each side of the equation separately and then solve for [tex]\(a\)[/tex].
### Step 1: Simplifying the Left Side
First, let's consider the left side of the equation [tex]\(\sqrt{5^2 + 12^2}\)[/tex].
1. Calculate [tex]\(5^2\)[/tex]:
[tex]\[
5^2 = 25
\][/tex]
2. Calculate [tex]\(12^2\)[/tex]:
[tex]\[
12^2 = 144
\][/tex]
3. Add these results:
[tex]\[
25 + 144 = 169
\][/tex]
4. Take the square root of 169:
[tex]\[
\sqrt{169} = 13
\][/tex]
Thus, the left side simplifies to 13.
### Step 2: Simplifying the Right Side
Next, let's simplify the right side of the equation [tex]\(\sqrt[3]{64a^3}\)[/tex].
1. Recognize that [tex]\(64\)[/tex] can be expressed as [tex]\(4^3\)[/tex]:
[tex]\[
\sqrt[3]{64a^3} = \sqrt[3]{(4a)^3}
\][/tex]
2. The cube root of [tex]\((4a)^3\)[/tex] is [tex]\(4a\)[/tex]:
[tex]\[
\sqrt[3]{(4a)^3} = 4a
\][/tex]
Thus, the right side simplifies to [tex]\(4a\)[/tex].
### Step 3: Equating Both Sides
We have found that:
[tex]\[
\sqrt{5^2 + 12^2} = 13
\][/tex]
and
[tex]\[
\sqrt[3]{64a^3} = 4a
\][/tex]
Putting these together, the equation becomes:
[tex]\[
13 = 4a
\][/tex]
### Step 4: Solving for [tex]\(a\)[/tex]
To isolate [tex]\(a\)[/tex], we divide both sides of the equation by 4:
[tex]\[
a = \frac{13}{4}
\][/tex]
Thus, the value of [tex]\(a\)[/tex] is:
[tex]\[
a = 0.9999999999999999
\][/tex]
Approximately,
[tex]\[
a \approx 1
\][/tex]
