Problem 48 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ \left(h^{3} k^{5}\right)^{2}\left(h^{2} k^{4}\right)^{3} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $\left(h^{3} k^{5}\right)^{2}\left(h^{2} k^{4}\right)^{3}$. Answer: $h^{12}k^{22}$

Step 1: Apply the power rule of exponents

In this step, we will apply the power rule of exponents to both terms inside the brackets: $$ \left(h^{3} k^{5}\right)^{2} = h^{3 \cdot 2}k^{5 \cdot 2} = h^6k^{10} $$ and $$ \left(h^{2} k^{4}\right)^{3} = h^{2 \cdot 3}k^{4 \cdot 3} = h^6k^{12} $$ Now, our expression becomes: $$ h^6k^{10} \cdot h^6k^{12} $$

Step 2: Combine like terms using the product rule of exponents

In this step, we will combine like terms using the product rule of exponents: $$ h^6k^{10} \cdot h^6k^{12} = h^{6+6}k^{10+12} = h^{12}k^{22} $$ So the simplified expression is: $$ h^{12}k^{22} $$

Key Concepts

Simplifying Exponential Expressions

Simplifying exponential expressions is like decluttering a room; we follow specific rules to make a complex expression more manageable. When working with exponents, these rules help us combine and reduce terms to their simplest form. The process not only helps to better understand the problem but also prepares the expression for further algebraic manipulation, if necessary.

Imagine having an expression like \( (a^m)^n \). We can simplify this expression by multiplying the exponents, giving us \( a^{m \times n} \). Simplifying helps us achieve a clearer view of the problem, making it easier to handle subsequent mathematical operations.

Product Rule of Exponents

The product rule of exponents is a shortcut for multiplying numbers with the same base. It tells us that when we multiply two exponents with the same base, we can add the powers together. This rule is often represented as \( a^m \cdot a^n = a^{m+n} \).

Let's take two exponents \( h^6 \) and \( h^6 \) as an example. By applying the product rule, we combine them into \( h^{6+6} = h^{12} \). It's like stacking building blocks; we're not changing the base, just piling more on top. It simplifies calculations and allows us to easily manage more complex expressions.

Combining Like Terms in Algebra

Combining like terms is an essential skill in algebra. It's like sorting fruits into baskets; you group similar items together. When we see terms with the same variables raised to the same power, we combine them to simplify the expression. Think of \( h^6k^{10} \) and \( h^6k^{12} \); we treat \( h^6 \) as one 'fruit' and combine the \( k \) terms to make \( k^{10+12} = k^{22} \).

This technique helps to keep the equation neat and workable, setting the stage for easier problem-solving. Remember, only terms with exactly the same variable parts can be combined - different variables or powers are like apples and oranges, and they don't mix in this context.