Problem 115 - Elementary Algebra

Question

Simplify the following problems. $$ \frac{\left(x^{4} y^{6} z^{10}\right)^{4}}{\left(x y^{5} z^{7}\right)^{3}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $\frac{(x^4y^6z^{10})^4}{(xy^5z^7)^3}$. Answer: The simplified expression is $x^{13}y^9z^{19}$.

Step 1: Apply the power of a power rule

Apply the power of a power rule to the numerator and the denominator: $$\left(x^{4}y^{6}z^{10}\right)^{4} = x^{4\times4}y^{6\times4}z^{10\times4} = x^{16}y^{24}z^{40}$$ $$\left(xy^{5}z^{7}\right)^{3} = x^{1\times3}y^{5\times3}z^{7\times3} = x^3y^{15}z^{21}$$ So the expression becomes: $$\frac{x^{16}y^{24}z^{40}}{x^3y^{15}z^{21}}$$

Step 2: Apply the quotient of powers rule

Apply the quotient of powers rule to each term: $$\frac{x^{16}}{x^3} = x^{16-3} = x^{13}$$ $$\frac{y^{24}}{y^{15}} = y^{24-15} = y^9$$ $$\frac{z^{40}}{z^{21}} = z^{40-21} = z^{19}$$ Now multiply the simplified terms together: $$x^{13} \cdot y^9 \cdot z^{19}$$

Step 3: Final Answer

The simplified expression is: $$x^{13}y^9z^{19}$$

Key Concepts

Exponent Rules

Exponent rules are essential tools in algebra for simplifying expressions. They dictate how to handle powers or exponents across various operations, helping us reduce expressions neatly.

Here are some core exponent rules:

• Product of Powers Rule: When multiplying similar bases, you add the exponents. \(a^m \times a^n = a^{m+n}\).
• Power of a Power Rule: When raising a power to another power, you multiply the exponents. \((a^m)^n = a^{m\times n}\).
• Quotient of Powers Rule: When dividing similar bases, you subtract the exponents. \(\frac{a^m}{a^n} = a^{m-n}\).
• Zero Exponent Rule: Any base raised to the power zero equals one. \(a^0 = 1\) (assuming \(a eq 0\)).
• Negative Exponent Rule: A negative exponent means you take the reciprocal of the base. \(a^{-n} = \frac{1}{a^n}\).

These rules are the foundation for simplifying expressions and solving complex algebraic problems, like the one in our exercise.

Power of a Power Rule

The Power of a Power Rule is a helpful shortcut when an exponential expression is raised to another power. Instead of having to expand and multiply everything out, this rule simplifies the process.

How It Works

Simply multiply the exponents together:

• For example, \((x^4)^3 = x^{4\times3} = x^{12}\).
• In the exercise, applying this rule to the numerator \((x^4y^6z^{10})^4\) leads to \(x^{16}y^{24}z^{40}\).
• Similarly, for the denominator \((xy^5z^7)^3\), we obtain \(x^3y^{15}z^{21}\).

This rule saves time and minimizes errors when simplifying expressions involving multiple powers.

Quotient of Powers Rule

The Quotient of Powers Rule is used when dividing exponential expressions with the same base. This rule helps to simplify the expression by reducing the exponents.

Application in Simplifying

To use the rule, subtract the exponent in the denominator from the exponent in the numerator:

• For example, \(\frac{x^5}{x^2} = x^{5-2} = x^3\).
• In our exercise, we simplify \(\frac{x^{16}}{x^3} = x^{13}\).
• The same goes for other variables: \(\frac{y^{24}}{y^{15}} = y^9\) and \(\frac{z^{40}}{z^{21}} = z^{19}\).

This makes the final simplified expression \(x^{13}y^9z^{19}\), using all simplified components together.