Problem 584 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\left(a^{6} b^{16}\right)^{\frac{1}{2}}\) (b) \(\left(j^{9} k^{6}\right)^{\frac{2}{3}}\)

Step-by-Step Solution

Answer

(a) a^3 b^8 (b) j^6 k^4

Step 1: Apply the power rule for exponents (a)

When simplifying \(\big(a^{m} b^{n}\big)^p\) apply the power rule which states \(\big(x^{a}\big)^{b} = x^{a \times b}\). For part (a) we get \(a^{6 \times \frac{1}{2}} b^{16 \times \frac{1}{2}}\).

Step 2: Simplify the exponents (a)

Simplify the exponents in the expression from Step 1: \(a^{6 \times \frac{1}{2}} = a^3\) and \(b^{16 \times \frac{1}{2}} = b^8\). So, the simplified form of (a) is \a^3 b^8\.

Step 3: Apply the power rule for exponents (b)

Similarly, for part (b), \(\big(j^9 k^6\big)^{\frac{2}{3}}\), use the power rule \(\big(x^{a}\big)^{b} = x^{a \times b}\). This gives \(j^{9 \times \frac{2}{3}} k^{6 \times \frac{2}{3}}\).

Step 4: Simplify the exponents (b)

Simplify the exponents in the expression from Step 3: \(j^{9 \times \frac{2}{3}} = j^6\) and \(k^{6 \times \frac{2}{3}} = k^4\). So, the simplified form of (b) is \j^6 k^4\.

Key Concepts

Power Rule for Exponents

The power rule for exponents is a helpful tool in algebra that simplifies expressions where an exponent is raised to another exponent. This rule is essential in many algebraic manipulations and can be summarized as:

• When you have a term of the form \(x^a\) raised to another power \(b\), the expression simplifies to \(x^{a \times b}\)\.

For example, if you need to simplify \((a^6 b^{16})^{1/2}\), you would apply the power rule to get:

• \(a^{6 \times 1/2} b^{16 \times 1/2}\)\.

Here, each exponent inside the parentheses is multiplied by the outside exponent of 1/2. This simplifies each term as follows:

• \(a^{6 \times 1/2} = a^3\), and
  \(b^{16 \times 1/2} = b^8\). Hence, the simplified form is \(a^3 b^8\)\.

Understanding and applying the power rule can save a lot of time and make complex expressions much more manageable.

Simplifying Exponents

Simplifying exponents involves reducing expressions with exponents to their simplest form. This is often done using rules like the power rule, product rule, and quotient rule. Consider the expression \((j^9 k^6)^{2/3}\):

• Apply the power rule to get \(j^{9 \times 2/3} k^{6 \times 2/3}\)\.

Next, you simplify each exponent:

• \(j^{9 \times 2/3} = j^6\), since \ 9 \times 2/3 \ simplifies to 6\.
• \(k^{6 \times 2/3} = k^4\), since \ 6 \times 2/3 \ simplifies to 4\.

Thus, the simplified form is \(j^6 k^4\)\. When simplifying exponents, it is critical to carefully multiply the exponent. Simplification ensures the expression is in its most compact and manageable form.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and operations (such as addition, subtraction, multiplication, division, and exponentiation). These expressions are the building blocks of algebra and are used to formulate and solve equations. An example of an algebraic expression is \((a^6 b^{16})^{1/2}\):

• This expression can be simplified by applying the power rule, resulting in \(a^3 b^8\)\.

Another example is \((j^9 k^6)^{2/3}\), which simplifies to \(j^6 k^4\)\:Understanding how to manipulate and simplify algebraic expressions is a core skill in algebra that facilitates solving more complex problems. Practice regularly, and use the rules of exponents to guide you through simplifications. Whether you're dealing with simple or complex expressions, the rules remain the same, making them straightforward once mastered.