Problem 40 - 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. $$ \left(x^{2} y^{3} z^{9} w^{7}\right)^{3} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is \(x^6 y^9 z^{27} w^{21}\).

Step 1: Identify the expression to be simplified

We are given the expression \(\left(x^{2} y^{3} z^{9} w^{7}\right)^{3}\) and need to simplify it using the power of a product rule.

Step 2: Apply the power of a product rule

To do this, we apply the rule which states that when raising a product of factors to a power, we raise each factor to that power separately. So, the expression becomes: \((x^2)^3 (y^3)^3 (z^9)^3 (w^7)^3\)

Step 3: Simplify the exponents

Now, we simplify each exponent by multiplying the inner exponent by the outer exponent: \(x^{(2 \times 3)} y^{(3 \times 3)} z^{(9 \times 3)} w^{(7 \times 3)}\) This results in the simplified expression: \(x^6 y^9 z^{27} w^{21}\)

Key Concepts

Exponents

In mathematics, exponents are a way to express repeated multiplication of a number by itself. For example, the expression \(x^a\) means that the base \(x\) is multiplied by itself \(a\) times. Here are some key ideas regarding exponents:

• The base is the number being multiplied.
• The exponent (or power) tells us how many times the base is used as a factor.
• An exponent of 1 means the base itself. For example, \(x^1 = x\).
• An exponent of 0 results in 1, assuming the base is not zero. This is because \(x^0 = 1\).

Please note that exponents follow specific rules that govern their operations, such as the multiplication and division of exponents, which we'll explore further in relation to simplifying expressions.

Power of a Product Rule

The power of a product rule is an essential law in exponents, making calculations easier when dealing with powers raised to higher powers. It states that when you have an entire product of multiple factors raised to the same power, you can apply the exponent to each factor individually. For a product like \((ab)^n\), it becomes \(a^n \times b^n\).

Here is a simple breakdown to comprehend:

• Identify the factors inside the parentheses.
• Raise each of them to the power outside the parentheses.

Applying this rule is particularly useful in simplifying large expressions efficiently and minimizing miscalculations. It allows you to address each component independently, as seen in the original exercise, where the whole expression \(\left(x^{2} y^{3} z^{9} w^{7}\right)^3\) splits into individual components raised to the third power.

Simplifying Expressions

To simplify mathematical expressions means to rewrite them in a more concise and manageable form without changing their value. When simplifying expressions with exponents, based on rules, you focus on aggregating the base elements and appropriately applying the operations to the exponents.

Steps to follow while simplifying expressions include:

• Apply the power of a product rule to separate components, as done in the provided solution.
• Multiply the exponents for each base according to the set mathematical operation. For instance, if you have \((x^a)^b\), it simplifies to \(x^{a \times b}\).
• Ensure the simplified form is accurate by rechecking each calculation, especially when dealing with multiple factors.

Simplifying expressions isn't just about finding the answer but also ensuring the equation is more understandable, which is particularly important when dealing with longer, more complex formulas.