Problem 39 - 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. $$ \left(a^{4} b^{7} c^{6} d^{8}\right)^{8} $$

Step-by-Step Solution

Answer

Answer: \(a^{32}b^{56}c^{48}d^{64}\)

Step 1: Recognize the expression and write the power rules

First, let's recognize that the given expression is in the form of a product raised to a power \((a^4 b^7 c^6 d^8)^8\). We shall use both the power of a product and the power of a power rules: - \((ab)^n = a^nb^n\) - \((a^m)^n = a^{mn}\)

Step 2: Apply the power of a product rule

Next, we will apply the power of a product rule to our given expression \((a^4 b^7 c^6 d^8)^8\). Here, we have four different terms inside the parentheses raised to a power: $$(a^4 b^7 c^6 d^8)^8 = (a^4)^8 (b^7)^8 (c^6)^8 (d^8)^8$$

Step 3: Apply the power of a power rule

Now, apply the power of a power rule to each term in the expression: - \((a^4)^8 = a^{4 \times 8} = a^{32}\) - \((b^7)^8 = b^{7 \times 8} = b^{56}\) - \((c^6)^8 = c^{6 \times 8} = c^{48}\) - \((d^8)^8 = d^{8 \times 8} = d^{64}\)

Step 4: Combine the simplified terms

Finally, write the simplified terms back together to form the final simplified expression: $$ a^{32}b^{56}c^{48}d^{64} $$ So, the given expression \(\left(a^{4} b^{7} c^{6} d^{8}\right)^{8}\) simplifies to \(a^{32}b^{56}c^{48}d^{64}\).

Key Concepts

Power Rules for Exponents

Understanding the power rules for exponents is crucial for simplifying expressions that contain exponents. The basic idea is that when you have an exponent raised to another exponent, you can multiply those exponents together. In mathematical terms, this rule is expressed as \( (a^m)^n = a^{m \times n} \). This principle allows us to combine powers in a step-by-step manner for ease of calculation.

Applying this to variables or numbers simplifies expressions significantly. For instance, if you have \( x^2 \) raised to the third power, using the power rule, it becomes \( x^{2 \times 3} \) or \(x^6\). It's a systematic approach that ensures expressions are simplified correctly without any guesswork.

Keep in mind that the bases must be the same when applying this rule. If the bases are different, you cannot combine the exponents. The power rule for exponents lays the foundation for working with more complex expressions and ensures that you have a clear pathway to simplification.

Power of a Product Rule

When you're dealing with a product (a set of numbers or variables multiplied together) that's raised to a power, the power of a product rule comes into play. This rule states that the power applies to each factor in the product independently: \( (ab)^n = a^n b^n \). If you have several factors, the same principle extends to them all \( (abc)^n = a^n b^n c^n \).

Using this rule, we can take an expression like \( (2xy)^3 \) and simplify it to \( 2^3 x^3 y^3 \) without altering the expression’s value. It is imperative to apply the power to both the numerical coefficients and the variables. This rule is particularly handy when simplifying algebraic expressions involving multiple terms with exponents. Through this rule, expressions become more manageable and prepare the groundwork for further algebraic manipulation or for plugging into equations.

Power of a Power Rule

The power of a power rule is yet another fundamental rule in the realm of exponents. This rule is applicable when an expression with an exponent is raised to another power, indicating that you multiply the exponents together. The general formula is \( (a^m)^n = a^{mn} \).

This rule helps in simplifying expressions like \( (x^2)^3 \) where you would multiply the exponents 2 and 3 to get \(x^6\). It's a straightforward concept, but it becomes invaluable when you encounter nested exponents, or expressions raised to higher powers.

Always remember that you only multiply the exponents when the bases are raised to a power, and not when they are multiplied together. This distinction is key and helps avoid common errors in simplification. The power of a power rule ensures that expressions like \( (a^4)^8 \) can be efficiently reduced to \(a^{32}\), streamlining the process of working with exponential expressions.