Problem 15 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rewrite the number without radicals or exponents.. $$ \left(\frac{27}{8}\right)^{2 / 3} $$

Step-by-Step Solution

Answer

The simplified expression without radicals or exponents is \(\frac{9}{4}\).

Step 1: Apply the exponent separately to the numerator and the denominator

Since we have a power of a fraction, we can apply the power of \(2/3\) separately to the numerator and the denominator: \[ \left(\frac{27}{8}\right)^{\frac{2}{3}}= \frac{27^{\frac{2}{3}}}{8^{\frac{2}{3}}}. \]

Step 2: Rewrite the exponent as a root expression and simplify

Now, we can rewrite the exponent as the cube root and then square the result. Notice that the cube root of 27 is 3 and the cube root of 8 is 2: \[ \frac{27^{\frac{2}{3}}}{8^{\frac{2}{3}}} = \frac{(\sqrt[3]{27})^2}{(\sqrt[3]{8})^2} = \frac{3^2}{2^2}. \]

Step 3: Complete the calculation

Finally, calculate the square of each number in the fraction: \[ \frac{3^2}{2^2} = \frac{9}{4}. \] As there are no more radicals or exponents, this is the simplified expression: \[ \left(\frac{27}{8}\right)^{\frac{2}{3}} = \frac{9}{4}. \]

Key Concepts

Exponents

Exponents are mathematical expressions that show how many times a number, known as the base, is multiplied by itself. For example, in the expression \(3^2\), 3 is the base and 2 is the exponent, meaning that 3 is multiplied by itself to give \(3 \times 3 = 9\).
Here are some key points to remember about exponents:

• An exponent like \(n^0\) is always 1, regardless of the base \(n\), unless the base is zero.
• Negative exponents indicate reciprocals. For example, \(x^{-2} = \frac{1}{x^2}\).
• Fractional exponents represent roots, such as \(x^{\frac{1}{2}} = \sqrt{x}\).
• Exponents manipulate the base by magnitude, like scaling numbers to huge or tiny values efficiently.

When dealing with algebra, understanding and applying exponent rules is essential for simplifying complex expressions.

Radicals

Radicals signify mathematical roots, with the most common being square roots and cube roots. The symbol for the square root is \(\sqrt{}\) and typically represents a power of \(\frac{1}{2}\). Cube roots, indicated by \(\sqrt[3]{}\), represent the exponent \(\frac{1}{3}\).
For example:

• \(\sqrt[3]{8} = 2\), because \(2 \times 2 \times 2 = 8\).
• \(\sqrt[3]{27} = 3\), as \(3 \times 3 \times 3 = 27\).

Understanding radicals is vital to simplify expressions containing exponents. Converting typically complex expressions into radicals helps identify base numbers and simplify these roots further for more manageable results.

Fraction Powers

Fraction powers emerge commonly in expressions involving radicals and exponents. They represent multiple operations: taking a root and raising to a power. For instance, the fraction power \(\frac{2}{3}\) means finding the cube root of a number and then squaring the result.
Let's clarify:

• Start with \(a^{\frac{m}{n}}\), which means \((\sqrt[n]{a})^m\).
• Applying \(27^{\frac{2}{3}}\) begins with finding the cube root of 27 (which is 3) and then consistent squaring (3 to the power of 2 equals 9).

This dual-operation of root extraction followed by known powers is crucial when simplifying expressions that initially appear demanding.

Simplification Process

Simplification is the process of reducing an expression to its simplest form, often by removing radicals or exponents. The key target is clarity and ease of calculation.
Steps to simplify expressions like \(\left(\frac{27}{8}\right)^{\frac{2}{3}}\) include:

• Separate the exponent to act on the numerator and denominator independently.
• Convert the fraction power to roots and exponentiation, i.e., cube root followed by squaring.
• Perform any calculations, such as \(3^2 = 9\) and \(2^2 = 4\), ending with a clear fraction \(\frac{9}{4}\).

Using logical steps transforms complex expressions into simplified forms, aiding in deeper comprehension and simpler computations.