$(($ {x^ $3$ })^{\frac{ $1$ }{ $2$ }}) = $4$ $

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Grade 6

Powers with decimal bases

Full solution

((

{x^

3

})^{\frac{

1

}{

2

}}) =

4

Simplify Left Side: We are given the equation $x^3$ ^( $1$ / $2$ ) = $4$ . We need to find the value of $x$ . To solve for $x$ , we will first simplify the left side of the equation using the power of a power rule, which states that $a^m$ ^n = a^{m*n}. In this case, we have $x^3$ ^( $1$ / $2$ ), which simplifies to $x^{3*(1/2)} = x^{3/2}$ .
Raise Both Sides: Now our equation is $x^{(3/2)} = 4$ . To solve for $x$ , we need to get rid of the exponent. We can do this by raising both sides of the equation to the reciprocal of $3/2$ , which is $2/3$ . We raise both sides of the equation to the power of $2/3$ : $(x^{(3/2)})^{(2/3)} = 4^{(2/3)}$ .
Calculate $4^{(2/3)}$ : When we raise a power to a power, we multiply the exponents. On the left side, we have $(3/2)*(2/3)$ which simplifies to $1$ , so we are left with $x^1$ or simply $x$ . On the right side, we need to calculate $4^{(2/3)}$ . We can break this down into two steps: first find the cube root of $4$ , and then square that result. The cube root of $4$ is not a whole number, but we can estimate it to be around $1.5874$ because $1.5874^3$ is approximately $4$ . Then we square $1.5874$ to get approximately $(3/2)*(2/3)$ $2$ . However, this is an estimation, and we need to recognize that $4^{(2/3)}$ is actually $(3/2)*(2/3)$ $4$ . Since $(3/2)*(2/3)$ $5$ is the cube root of $(3/2)*(2/3)$ $6$ , and $(3/2)*(2/3)$ $6$ is $(3/2)*(2/3)$ $8$ , we take the cube root of $(3/2)*(2/3)$ $8$ to get the exact value, which is $1$ $0$ .