Question

1. the length of the side of a cube is 5³ units. express its surface area (sa) and volume (v) using powers and simplify each expression.

Explanation:

Step1: Recall the formula for the surface area of a cube

The surface area formula of a cube is \(SA = 6s^{2}\), where \(s\) is the side - length of the cube.

Step2: Substitute \(s = 5^{3}\) into the formula

Substitute \(s\) into the formula: \(SA=6\times(5^{3})^{2}\).

Step3: Simplify the exponent using the power - of - a - power rule \((a^{m})^{n}=a^{mn}\)

For \((5^{3})^{2}\), by the power - of - a - power rule \(a = 5\), \(m = 3\), \(n = 2\), so \((5^{3})^{2}=5^{3\times2}=5^{6}\).

Step4: Calculate the surface area

Then \(SA = 6\times5^{6}\).

Step5: Recall the formula for the volume of a cube

The volume formula of a cube is \(V=s^{3}\), where \(s\) is the side - length of the cube.

Step6: Substitute \(s = 5^{3}\) into the volume formula

Substitute \(s\) into the formula: \(V=(5^{3})^{3}\).

Step7: Simplify the exponent using the power - of - a - power rule \((a^{m})^{n}=a^{mn}\)

For \((5^{3})^{3}\), by the power - of - a - power rule \(a = 5\), \(m = 3\), \(n = 3\), so \((5^{3})^{3}=5^{3\times3}=5^{9}\).

Answer:

The surface area \(SA = 6\times5^{6}\) and the volume \(V = 5^{9}\)