To solve the expression [tex]\( 64^{1/3} \)[/tex], we need to find the cube root of 64. This can be understood as finding a number [tex]\( x \)[/tex] such that [tex]\( x^3 = 64 \)[/tex].
First, let's recall that the cube root of a number is the value that, when multiplied by itself three times, gives the original number. Symbolically, for any number [tex]\( a \)[/tex]:
[tex]\[ a^{1/3} = x \implies x^3 = a \][/tex]
We are given the expression [tex]\( 64^{1/3} \)[/tex]. Thus, we need to determine:
[tex]\[ 64^{1/3} = x \][/tex]
such that:
[tex]\[ x^3 = 64 \][/tex]
Let's test possible values:
1. [tex]\( 4 \)[/tex]:
[tex]\[ 4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64 \][/tex]
So, [tex]\( 4^3 = 64 \)[/tex]. This confirms that [tex]\( x = 4 \)[/tex] is the correct value.
Now, let's evaluate other choices:
2. [tex]\( \frac{64}{3} \)[/tex]:
[tex]\[ \left( \frac{64}{3} \right)^3 \][/tex]
This value is not an integer, and its cube would not equal 64.
3. [tex]\( 8 \)[/tex]:
[tex]\[ 8^3 = 8 \times 8 \times 8 = 64 \times 8 = 512 \][/tex]
This is not equal to 64.
4. [tex]\( \frac{4}{3} \)[/tex]:
[tex]\[ \left( \frac{4}{3} \right)^3 = \frac{4^3}{3^3} = \frac{64}{27} \][/tex]
This is not equal to 64.
Thus, we find that the value which equates to [tex]\( 64^{1/3} \)[/tex] is indeed [tex]\( 4 \)[/tex].
Therefore, the solution to the expression is:
[tex]\[ 64^{1/3} = 4 \][/tex]
The correct answer is:
A. 4
