Question

Simplify: \(\left(\frac{4}{25}\right)^{3 / 2}\)

Short answer

\(\frac{8}{125}\)

Step-by-step solution

Understand the Expression

The given expression is \(\frac{4}{25})^{3 / 2}\). It is a fractional exponent.

Break Down the Exponent

The exponent \(\frac{3}{2}\) can be split into two parts: \(\frac{3}{2} = 1.5 = 1 + 0.5\). This can be interpreted as raising to the power of 1 and then taking the square root.

Raise the Fraction to the First Power

Raising the fraction to the first power, we get \(\frac{4}{25}^1 = \frac{4}{25}\).

Take the Square Root of the Fraction

Next, find the square root of \(\frac{4}{25}\). This is \(\frac{\text{sqrt}(4)}{\text{sqrt}(25)} = \frac{2}{5}\).

Raise the Result to the Remaining Power

We need to raise the result (\frac{2}{5}) to the power of 3: \(\frac{2}{5}^3 = \frac{2^3}{5^3} = \frac{8}{125}\).

Key concepts

simplifying expressions

Simplifying expressions means making them easier to work with, especially by reducing complexity. In the exercise given, we start with \(\left(\frac{4}{25}\right)^{3 / 2}\). The goal is to transform this into a simpler form. We can do this by breaking down the steps, ensuring each operation is clear. This usually involves operations such as factoring, combining like terms, and using properties of exponents and roots. By simplifying, not only does the expression look cleaner, but it becomes more manageable and easier to understand or further manipulate.

exponentiation

Exponentiation is a mathematical operation that involves raising a number to a power. For example, in \(x^y\), 'x' is the base and 'y' is the exponent. It tells us how many times to multiply the base by itself. In our exercise, we deal with fractional exponents, specifically \(\left(\frac{4}{25}\right)^{3 / 2}\). A fractional exponent combines both an exponent and a root. Here, \(\frac{3}{2}\) implies raising to the power of 3 and taking the square root. Exponentiation helps us handle large numbers or simplifying complex expressions.

square roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \(3 \times 3 = 9\). In our problem, the square root comes into play with the exponent \(\left(\frac{3}{2}\right)\), which we can split into two parts: 1 and \(\frac{1}{2}\). Taking the square root of a fraction like \(\frac{4}{25}\) is done by finding the square roots of the numerator and the denominator separately. So, \(\frac{\text{sqrt}(4)}{\text{sqrt}(25)} = \frac{2}{5}\).

fractions

Fractions represent a part of a whole and are written as two numbers separated by a slash. The top number is the numerator, and the bottom is the denominator. In the exercise, \(\left(\frac{4}{25}\right)^{3 / 2}\) is a fraction: 4 is the numerator, and 25 is the denominator. When dealing with fractions in exponentiation, each part of the fraction undergoes the operations. For instance, to simplify \(\left(\frac{4}{25}\right)^{3 / 2}\), we handle the fraction step-by-step—first raising it to a power, then taking roots, and then raising it again if needed.