What is 3/4 to the Third Power? And please show how to reach solution

What is 3/4 to the Third Power?

To find (3/4) to the third power, you raise 3/4 to the exponent 3:

(3/4)^3 = (3/4) * (3/4) * (3/4) = 27/64.

So, (3/4) to the third power is equal to 27/64. You obtain this by cubing both the numerator (3) and denominator (4) separately and simplifying the fraction.

Raising a Fraction to a Power

When given an expression with a fraction raised to a power like (3/4)^3, we evaluate it by raising the numerator and denominator separately to that power. Let’s walk through the process for calculating (3/4)^3.

Understanding Exponents

An exponent indicates repeated multiplication of the base number.

For example:

53 = 5 * 5 * 5 = 125

The exponent 3 means to multiply the base 5 by itself 3 times.

This works with fractions too:

(3/4)^3 means to take 3/4 and multiply it by itself 3 times.

Raising the Numerator and Denominator

To raise a fraction to an exponent:

1. Raise the numerator to that power
2. Raise the denominator to that power

For (3/4)^3:

Numerator: 3^3 = 27 Denominator: 4^3 = 64

So (3/4)^3 = 27/64

Why It Works

This method maintains the part-to-whole relationship of the original fraction:

(3/4) * (3/4) * (3/4) = 27/64

Multiplying the numerator and denominator separately still gives the result of multiplying (3/4) three times.

Checking the Work

We can verify the result by entering the original expression into a calculator, which will give 27/64 for (3/4)^3. Getting the same result confirms the manual calculation.

Being able to raise fractions to powers lays a solid foundation for more complex math. Keeping the numerator and denominator separate maintains the meaning of exponents for fractional bases.

In summary, to raise a fraction to an exponent, raise the numerator and denominator separately to that power while keeping them over each other.