Problem 7 - Contemporary Precalculus

Question

Simplify the expression. Assume \(a, b, c, d>0\) $$\frac{(7 a)^{2}(5 b)^{3 / 2}}{(5 a)^{3 / 2}(7 b)^{4}}$$

Step-by-Step Solution

Answer

Question: Simplify the expression: $$\frac{(7 a)^{2}(5 b)^{3 / 2}}{(5 a)^{3 / 2}(7 b)^{4}}$$, where \(a, b, c, d > 0\). Answer: $$\frac{a^{\frac{1}{2}}}{49b^{\frac{5}{2}}}$$

Step 1: Rewrite the expression in terms of powers

We will rewrite the expression in terms of powers such that it becomes easier to simplify. The given expression is: $$\frac{(7 a)^{2}(5 b)^{3 / 2}}{(5 a)^{3 / 2}(7 b)^{4}}$$ Using the power rule (multiplication) \( (x^m)^n = x^{mn},\) we rewrite the expression as: $$\frac{7^{2}a^{2} \cdot 5^{\frac{3} {2}}b^{\frac{3} {2}}}{5^{\frac{3} {2}}a^{\frac{3}{2}} \cdot 7^{4} b^{4}}$$

Step 2: Simplify exponential terms by utilizing exponent rules

We simplify exponential terms using the quotient rule of exponents (division): \(x^m/x^n = x^{m-n}\). Our expression now becomes: $$\frac{7^{2-4}a^{2-\frac{3}{2}} \cdot 5^{\frac{3}{2}-\frac{3}{2}}b^{\frac{3}{2}-4}}{1}$$ $$= 7^{-2}a^{\frac{1}{2}} \cdot 5^{0}b^{-\frac{5}{2}}$$

Step 3: Utilize exponential rules for negative exponents

Next, we use the negative exponent rule: \(x^{-n} = \frac{1}{x^n}\). This gives us: $$\frac{a^{\frac{1}{2}}}{7^2} \cdot \frac{5^0}{b^{\frac{5}{2}}}$$

Step 4: Simplify and write the final expression

Now we simplify the constant terms, noting that \(5^0 = 1\) and combine the remaining terms to get the final simplified expression: $$\frac{a^{\frac{1}{2}}}{49b^{\frac{5}{2}}}$$

Key Concepts

Exponent Rules

Understanding exponent rules is crucial when simplifying exponential expressions. These rules provide a systematic method to manipulate expressions involving powers. One fundamental rule is the power rule, which allows you to multiply exponents when raising an exponent to another power, as seen with \( (x^m)^n = x^{mn} \). When dividing exponents with the same base, the quotient rule comes into play: \(x^m / x^n = x^{m-n}\), effectively subtracting the exponents. Lastly, the negative exponent rule is particularly helpful to simplify terms with negative powers: \(x^{-n} = 1/x^n\), meaning you take the reciprocal of the base raised to the positive power. In the provided exercise, these rules are applied successively to reach the simplified form of the given expression.

For instance, \(7^{2} / 7^{4}\) becomes \(7^{-2}\) using the quotient rule. Then, according to the negative exponent rule, \(7^{-2}\) transforms into \(1/7^{2}\). This process systematically reduces complex expressions into simpler ones that are easier to understand and calculate.

Radicals and Rational Exponents

Radicals and rational exponents are two sides of the same coin. A radical expression can be rewritten as an expression with a rational exponent and vice versa. This conversion is beneficial for simplification. The \(n\)th root of a number \(x\) is equivalent to raising \(x\) to the power of \(1/n\), symbolized as \(\sqrt[n]{x} = x^{1/n}\).

Conversion between Radicals and Rational Exponents

When simplifying, converting to a rational exponent can make it easier to apply the quotient rule for exponents. In the exercise, \(5b)^{3 / 2}\) represents the square root of \(5b\) cubed, which could also be written as \(\sqrt[2]{(5b)^3}\). Similarly, the term \(a^{\frac{1}{2}}\) signifies the square root of \(a\). These conversions are applied to consolidate terms with fractional exponents, ultimately leading to a more straightforward expression.

Algebraic Simplification

The goal of algebraic simplification is to present expressions in their most reduced and easily interpretable form. After applying the exponent rules and understanding the interplay between radicals and rational exponents, algebraic simplification involves combining like terms and reducing fractions whenever possible. In the provided exercise, this involved recognizing that \(5^{\frac{3}{2} - \frac{3}{2}} = 5^{0} = 1\) and thus eliminating that term from the equation.

Final Steps in Simplification

Towards the end of the process, we also simplified \(7^{2}\) to \(49\) and rewrote \(b^{-\frac{5}{2}}\) as \(1 / b^{\frac{5}{2}}\). By performing these steps, the original complex expression was reduced to \(\frac{a^{\frac{1}{2}}}{49b^{\frac{5}{2}}}\), which is much simpler to interpret. Through algebraic simplification, students can achieve clarity and ease in understanding otherwise daunting expressions.