Problem 35
Question
\(\left(\frac{h}{k}\right)^{7}\)
Step-by-Step Solution
Verified Answer
The simplified expression is \(\frac{h^7}{k^7}\).
1Step 1: Understand the Exponentiation
Identify that \(\frac{h}{k}\right)^{7}\) indicates the fraction \(\frac{h}{k}\) being raised to the power of 7.
2Step 2: Distribute the Exponent
Apply the exponent to both the numerator and denominator separately: \[ \frac{h^7}{k^7} \]
3Step 3: Write the Final Expression
The simplified form of \left( \frac{h}{k} \right)^{7} is \(\frac{h^7}{k^7}\).
Key Concepts
Raising Fractions to a PowerExponent RulesSimplifying Expressions
Raising Fractions to a Power
When we raise a fraction to a power, we are applying the exponent to both the numerator and the denominator of the fraction.
For example, \(\frac{h}{k}\)^{7} means we apply the exponent 7 to both \(h\) (the numerator) and \(k\) (the denominator).
So, this becomes \[ \frac{h^7}{k^7} \] Remember, the exponent affects every term inside the fraction. If there were more terms in the numerator or denominator, each term would be raised to the given power.
This is a fundamental rule of exponents for fractions and it helps simplify the expression correctly.
For example, \(\frac{h}{k}\)^{7} means we apply the exponent 7 to both \(h\) (the numerator) and \(k\) (the denominator).
So, this becomes \[ \frac{h^7}{k^7} \] Remember, the exponent affects every term inside the fraction. If there were more terms in the numerator or denominator, each term would be raised to the given power.
This is a fundamental rule of exponents for fractions and it helps simplify the expression correctly.
Exponent Rules
Exponent rules help us manage and simplify expressions involving powers. Here are some essential rules:
\textbf{1. Product of Powers Rule}: When you multiply two exponents with the same base, you add the exponents.\[ a^m \times a^n = a^{m+n} \]
\textbf{2. Quotient of Powers Rule}: When you divide two exponents with the same base, you subtract the exponents.\[ \frac{a^m}{a^n} = a^{m-n} \text{ where } a eq 0 \]
\textbf{3. Power of a Power Rule}: When you raise an exponent to another power, you multiply the exponents.\[ (a^m)^n = a^{mn} \]
\textbf{4. Power of a Product Rule}: When you raise a product to a power, you apply the exponent to each factor inside the product.\[ (ab)^m = a^m b^m \]
\textbf{5. Power of a Quotient Rule}: Similar to the above, when you raise a fraction to a power, you apply the exponent to both the numerator and the denominator.\[ \bigg(\frac{a}{b}\bigg)^m = \frac{a^m}{b^m} \text{ where } b eq 0 \]
Understanding these rules helps in simplifying and manipulating expressions involving exponents efficiently.
\textbf{1. Product of Powers Rule}: When you multiply two exponents with the same base, you add the exponents.\[ a^m \times a^n = a^{m+n} \]
\textbf{2. Quotient of Powers Rule}: When you divide two exponents with the same base, you subtract the exponents.\[ \frac{a^m}{a^n} = a^{m-n} \text{ where } a eq 0 \]
\textbf{3. Power of a Power Rule}: When you raise an exponent to another power, you multiply the exponents.\[ (a^m)^n = a^{mn} \]
\textbf{4. Power of a Product Rule}: When you raise a product to a power, you apply the exponent to each factor inside the product.\[ (ab)^m = a^m b^m \]
\textbf{5. Power of a Quotient Rule}: Similar to the above, when you raise a fraction to a power, you apply the exponent to both the numerator and the denominator.\[ \bigg(\frac{a}{b}\bigg)^m = \frac{a^m}{b^m} \text{ where } b eq 0 \]
Understanding these rules helps in simplifying and manipulating expressions involving exponents efficiently.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form so they are easier to understand and work with. Here are some tips for simplifying expressions with exponents:
\textbf{1. Combine Like Terms}: Collect similar terms to simplify an expression. For instance, \[ 3a^2 + 4a^2 = 7a^2\]
\textbf{2. Factor Using GCD}: Use the Greatest Common Divisor to factor out common elements. For example, \[ 12a^3 + 18a^2 = 6a^2(2a + 3) \]
\textbf{3. Apply Exponent Rules}: Use the rules of exponents to simplify complex expressions. E.g., simplify \[ (x^2)^3 \times \frac{x^5}{x^3} = x^{6-3+5} = x^8. \]
\textbf{4. Cancel Out Terms}: Cancel terms in a fraction to simplify.\[ \frac{4a^3b}{2ab} = \frac{4a^2b}{2b} = 2a^2 \]
By applying these strategies, you can make expressions simpler to work with. Simplification helps in solving and understanding mathematical problems more efficiently.
\textbf{1. Combine Like Terms}: Collect similar terms to simplify an expression. For instance, \[ 3a^2 + 4a^2 = 7a^2\]
\textbf{2. Factor Using GCD}: Use the Greatest Common Divisor to factor out common elements. For example, \[ 12a^3 + 18a^2 = 6a^2(2a + 3) \]
\textbf{3. Apply Exponent Rules}: Use the rules of exponents to simplify complex expressions. E.g., simplify \[ (x^2)^3 \times \frac{x^5}{x^3} = x^{6-3+5} = x^8. \]
\textbf{4. Cancel Out Terms}: Cancel terms in a fraction to simplify.\[ \frac{4a^3b}{2ab} = \frac{4a^2b}{2b} = 2a^2 \]
By applying these strategies, you can make expressions simpler to work with. Simplification helps in solving and understanding mathematical problems more efficiently.