Problem 49 - 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. $$ \left(x^{4} y^{3} z\right)^{4}\left(x^{5} y z^{2}\right)^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $(x^{4} y^{3} z)^{4}(x^{5} y z^{2})^{2}$. Answer: $x^{26} y^{14} z^{8}$

Step 1: Apply the exponent rules to each term within both parentheses

We start by applying the power rules to each term within the parentheses separately. In this step we use the rule \((a^m)^n = a^{mn}\), which means to multiply the exponents when you have an exponent raised to another power. $$ \left(x^{4} y^{3} z\right)^{4} = x^{4\cdot 4} y^{3\cdot 4} z^{1\cdot 4} \quad and \quad \left(x^{5} y z^{2}\right)^{2} = x^{5\cdot 2} y^{1\cdot 2} z^{2\cdot 2} $$

Step 2: Simplify the terms

Next, we will simplify the expressions from the previous step by multiplying the exponents. $$ \left(x^{4} y^{3} z\right)^{4} = x^{16} y^{12} z^{4} \quad and \quad \left(x^{5} y z^{2}\right)^{2} = x^{10} y^{2} z^{4} $$

Step 3: Multiply the two simplified expressions

Now, we will multiply the two simplified expressions obtained in step 2 by combining the like terms. We use the rule \(a^m \cdot a^n = a^{m+n}\), which means to add the exponents when you multiply terms with the same base. $$ (x^{16} y^{12} z^{4})(x^{10} y^{2} z^{4}) = x^{16+10} y^{12+2} z^{4+4} $$

Step 4: Simplify the final expression

Finally, we will simplify the expression obtained in step 3 by adding the exponents of like terms. $$ x^{16+10} y^{12+2} z^{4+4} = x^{26} y^{14} z^{8} $$ The simplified expression for the given problem is: $$ (x^{4} y^{3} z)^{4}(x^{5} y z^{2})^{2} = x^{26} y^{14} z^{8} $$

Key Concepts

Power of a Power Rule

When you see an exponent raised to another exponent, it might seem complex at first, but it’s actually quite simple. The power of a power rule states that \((a^m)^n = a^{m \cdot n}\). This means if you have something like \((x^4)^3\), you multiply 4 and 3 together, giving you \(x^{12}\).

So, whenever you encounter an exponent on top of an exponent, just remember this rule: multiply the exponents.

• Apply to each term inside the parentheses.
• Simplify by multiplying the exponents.

Using this, you can handle complex expressions efficiently!

Multiplying Exponents

Once you simplify each part using the power of a power rule, you might end up with different bases having exponents. To combine them, you use the multiplying exponents rule, which says \(a^m \cdot a^n = a^{m+n}\).

Notice that the bases must be the same. For example:

• \(x^3 \cdot x^4 = x^{3+4} = x^7\)
• Combine terms with identical bases by adding the exponents.

This helps in consolidating expressions to a single term with a powerful exponent.

Natural Numbers

Natural numbers are simply the familiar counting numbers starting from 1, 2, 3, and so on. In the world of exponents, when you see that exponents are natural numbers, it just means that these will always be positive integers, not fractions or negatives.

Natural numbers tie into the rules of exponents very smoothly:

• Exponents as natural numbers ensure straightforward calculations.
• Using positive exponents means expressions grow bigger rather than shifting into fractions.

Thus, knowing the exponents are natural numbers makes working with them easier!

Like Terms

In algebra, like terms are terms that have the same variables raised to the same powers. They play a key role when simplifying expressions.

For instance, if you have multiple terms like \(x^2\) and \(x^2\), they can be combined because they are like terms. When multiplying such terms, you'll use techniques such as adding their exponents.

• Ensure terms have identical bases and powers.
• Use like terms to simplify complex expressions efficiently.

Recognizing and correctly combining like terms is crucial for simplifying and solving algebraic equations.