Problem 570 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\left(m^{6}\right)^{\frac{5}{2}}\) (b) \(\left(n^{9}\right)^{\frac{4}{3}}\) (c) \(\left(p^{12}\right)^{\frac{3}{4}}\)

Step-by-Step Solution

Answer

(a) \m^{15}\, (b) \^{12}\, (c) \p^9\

Step 1: Use the Power Rule for Exponents

For this exercise, use the power rule for exponents which states that \(\big(a^m\big)^n = a^{m \times n}\).

Step 2: Simplify Part (a)

Apply the power rule: \(\big(m^6\big)^{\frac{5}{2}} = m^{6 \times \frac{5}{2}}\). Now calculate the exponent: \6 \times \frac{5}{2} = 15\, so the expression simplifies to \m^{15}\.

Step 3: Simplify Part (b)

Apply the power rule: \(\big(n^9\big)^{\frac{4}{3}} = n^{9 \times \frac{4}{3}}\). Now calculate the exponent: \9 \times \frac{4}{3} = 12\, so the expression simplifies to \^{12}\.

Step 4: Simplify Part (c)

Apply the power rule: \(\big(p^{12}\big)^{\frac{3}{4}} = p^{12 \times \frac{3}{4}}\). Now calculate the exponent: \12 \times \frac{3}{4} = 9\, so the expression simplifies to \p^9\.

Key Concepts

Power Rule for Exponents

The Power Rule for Exponents is a fundamental concept in algebra that simplifies dealing with powers raised to powers. According to this rule, if you have an expression in the form \(\big(a^m\big)^n\), you can write it as \({a}^{m \times n}\). This simplifies the process significantly. Let's take a closer look at how this works with some examples from the exercise. When you have \(\big(m^6\big)^{\frac{5}{2}}\), you can reframe it using the power rule: \(\big(m^6\big)^{\frac{5}{2}} = m^{6 \times \frac{5}{2}}\). When you carry out the multiplication, 6 times \(\frac{5}{2}\) results in 15, so the simplified form is \({m}^{15}\).The same process applies for other exponents, such as \(\big(n^9\big)^{\frac{4}{3}}\). Using the power rule, you obtain \({n}^{9 \times \frac{4}{3}}\), which simplifies to \({n}^{12}\), and \(\big(p^{12}\big)^{\frac{3}{4}}\) simplifies to \({p}^{12 \times \frac{3}{4}} = p^9\).

Simplification

Simplification is key in algebra. It means making expressions as concise and manageable as possible. When dealing with exponents, simplification often involves using rules like the Power Rule for Exponents. For instance, let's take the expression \(\big(m^6\big)^{\frac{5}{2}}\). Starting with the power rule, it becomes \({m}^{6 \times \frac{5}{2}}\), and multiplying the exponents, you get \({m}^{15}\). This streamlined form is much easier to work with. Simplifying expressions helps avoid errors in further calculations. Applying the same method to \(\big(n^9\big)^{\frac{4}{3}}\) turns it into \({n}^{12}\), and likewise, \(\big(p^{12}\big)^{\frac{3}{4}}\) simplifies to \({p}^{9}\). Always remember to simplify your exponents when possible. It saves time and reduces complexity in subsequent steps.

Algebra

Algebra involves solving equations and manipulating expressions involving variables. Understanding and applying exponent rules, like the Power Rule, is crucial for simplifying and solving algebraic expressions. Consider the expression \(\big(m^6\big)^{\frac{5}{2}}\). Using the Power Rule for Exponents allows you to rewrite and simplify it to \({m}^{15}\). This is much easier to interpret and use in further algebraic manipulations. The same approach can be applied to simplifying other expressions in the exercise, such as \(\big(n^9\big)^{\frac{4}{3}}\), which becomes \({n}^{12}\), and \(\big(p^{12}\big)^{\frac{3}{4}}\), which becomes \({p}^{9}\). These simplifications make algebraic equations less daunting and more straightforward to solve. Mastering these techniques will improve your efficiency and accuracy in algebra.