Problem 27 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ \left(\frac{25}{36}\right)^{3 / 2} $$

Step-by-Step Solution

Answer

The value of \(\left(\frac{25}{36}\right)^{3/2}\) is \(\frac{125}{216}\).

Step 1: Understand the Expression

The expression given is \( \left(\frac{25}{36}\right)^{3 / 2} \). This is a fractional exponent, which involves both exponentiation and taking roots of the fraction.

Step 2: Apply Fractional Exponent Rule

The expression \((a/b)^{m/n}\) can be written as \(\sqrt[n]{(a/b)^m}\). Here, \(n=2\) and \(m=3\), so we rewrite the expression as \(\sqrt{\left(\frac{25}{36}\right)^3}\). First, we'll find the square root of \(\frac{25}{36}\).

Step 3: Calculate the Square Root of the Fraction

Calculate the square root of both the numerator and the denominator separately. The square root of 25 is 5, and the square root of 36 is 6. Therefore, \(\sqrt{\frac{25}{36}} = \frac{5}{6}\).

Step 4: Raise the Result to the Power of 3

Now, raise \(\frac{5}{6}\) to the third power. To do this, raise both the numerator and the denominator to the power of 3. The cube of 5 is 125, and the cube of 6 is 216. Thus, \(\left(\frac{5}{6}\right)^3 = \frac{125}{216}\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of another number, called the exponent. This operation is commonly used to express repeated multiplication. For example, when we say \( 2^3 \), it means multiplying 2 by itself 3 times, which equals 8.

In the context of fractional exponents, exponentiation becomes a bit more complex. Here, the exponent is a fraction rather than a whole number. For instance, in the expression \( (a/b)^{m/n} \), both the base \( a/b \) and the exponent \( m/n \) are involved in the calculation. This specific type of exponentiation requires you to think about two operations: raising the quotient to the power \( m \) and taking the \( n \)-th root of the result.

This method allows us to handle more complicated expressions without the need for a calculator, by breaking it down into smaller and more manageable calculations.

Square Roots

Square roots are a fundamental concept in algebra, representing one of the two equal factors of a number. For example, the square root of 25 is 5, because \( 5 \times 5 = 25 \). In general, taking the square root of a number is the inverse operation of squaring that number.

When dealing with fractional exponents such as \( \left(rac{a}{b}\right)^{1/2} \), we handle each part of the fraction separately. The square root of a fraction involves calculating the square root of both the numerator and the denominator independently. For instance, \( rac{25}{36} \) transforms into \(rac{\sqrt{25} \}{\sqrt{36}}\), resulting in \(rac{5}{6} \).

This separate computation helps to simplify the expression into a more straightforward fraction without the use of a calculator, making it a crucial operation when working with expressions involving fractional exponents.

Fraction Calculation

Fraction calculation involves performing arithmetic operations with fractions, such as addition, subtraction, multiplication, and division. The key to working with fractions is knowing how to manipulate both the numerators and the denominators properly.

In the context of fractional exponents like \(rac{5}{6} \), we initially dealt with finding square roots, giving us a simpler fraction. Once simplified, further operations can be performed – such as raising the fraction to a power – by handling the numerators and denominators separately.

For example, taking \( rac{5}{6} \) to the power of 3 involves separately cubing the numerator and the denominator. The cube of 5 is 125, and the cube of 6 is 216, leading to the fraction \( rac{125}{216} \). This process requires careful calculation but can be easily managed by breaking it into smaller, more straightforward steps.