Problem 79 - Algebra for College Students

Question

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(\frac{60 a^{\frac{1}{5}}}{15 a^{\frac{3}{4}}}\right)^{2}\)

Step-by-Step Solution

Answer

\(\frac{16}{a^{\frac{11}{10}}}\).

Step 1: Simplify the Fraction

First, simplify the base of the expression by dividing the numerical coefficients and the variable parts separately. - Divide the coefficients: \(\frac{60}{15} = 4\).- For the variables, use the rule \(a^m / a^n = a^{m-n}\): \(a^{\frac{1}{5}} / a^{\frac{3}{4}} = a^{\frac{1}{5} - \frac{3}{4}} = a^{-\frac{11}{20}}\). Thus, the expression becomes \( (4 a^{-\frac{11}{20}})^2 \).

Step 2: Apply the Power to the Fraction

Raise each term in the fraction to the power of 2. - For the coefficient: \((4)^2 = 16\).- For the variable with exponent: Use the rule \((a^m)^n = a^{m \cdot n}\): \((a^{-\frac{11}{20}})^2 = a^{-(\frac{11}{20} \times 2)} = a^{-\frac{11}{10}}\). The expression is now \(16 a^{-\frac{11}{10}}\).

Step 3: Convert Negative Exponent to Positive

Finally, express the answer with positive exponents only by moving the term with the negative exponent to the denominator:- The expression \(a^{-\frac{11}{10}}\) can be rewritten as \(\frac{1}{a^{\frac{11}{10}}}\).Thus, the final simplified expression is \(\frac{16}{a^{\frac{11}{10}}}\).

Key Concepts

fraction simplification

Simplifying fractions in algebra involves both the numerical and variable components of an expression.
To simplify a fraction, break it down into more manageable parts:

• For the numerical part, divide the coefficients (numbers without variables). This is straightforward arithmetic simplification. For example, dividing 60 by 15 gives 4.


• For variables, use the law of exponents: \(a^m / a^n = a^{m-n}\). This rule helps to simplify expressions involving the same base raised to different powers. In our example, dividing \(a^{\frac{1}{5}}\) by \(a^{\frac{3}{4}}\) results in \(a^{\frac{1}{5} - \frac{3}{4}} = a^{-\frac{11}{20}}\).

The goal is to use as many principles of basic arithmetic and exponent rules as possible. Simplifying the fraction this way forms the foundation for the rest of the problem.

negative exponents

Negative exponents might seem tricky at first, but they are actually straightforward when you break them down.
A negative exponent means you take the reciprocal of the base raised to the positive version of the exponent.

• This means \(a^{-n} = \frac{1}{a^n}\). If you have a negative exponent in a fraction, you move the base to the opposite part of the fraction (numerator to denominator or vice versa).


• In our problem, \((a^{-\frac{11}{20}})^2\) needs the exponent to be managed before simplifying further. You apply \((a^m)^n = a^{m \cdot n}\) to get \(a^{-\frac{11}{10}}\).

Understanding how negative exponents operate helps simplify expressions and ultimately express them in the most simplified form with positive exponents.

algebraic expressions

Algebraic expressions combine numbers and variables into manageable equations or expressions.
They often incorporate multiple arithmetic operations, including exponentiation, which requires understanding properties of exponents to solve.

• When dealing with exponents in algebraic expressions, the key is to apply laws such as the product of powers \(a^m \cdot a^n = a^{m+n}\), quotient of powers \(a^m / a^n = a^{m-n}\), and power of a power \((a^m)^n = a^{m \cdot n}\).


• The goal is not just to compute the expressions but also to simplify them for further use or integration into larger equations.


• In algebra, achieving a simplified state with positive exponents often makes expressions easier to interpret and apply in real-world situations.

Mastery of algebraic expressions allows for more efficient problem-solving and deeper understanding of mathematical relationships.