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Magazine Chapter 5: Problem 105
Simplify each expression. $$ \frac{\left(2 a^{5} b^{3}\right)^{4}}{-16 a^{20} b^{7}} $$
Short Answer
Expert verified
The simplified expression is \(-b^5\).
Step by step solution
01
Apply the Power of a Power Rule
First, let's simplify the numerator by applying the rule \((x^m)^n = x^{m\cdot n}\). For \((2a^5b^3)^4\), each component inside the parentheses is raised to the fourth power, so we have: \(2^4 \cdot (a^5)^4 \cdot (b^3)^4\). This simplifies to \(16a^{20}b^{12}\).
02
Simplify the Fraction
Now, substitute the simplified numerator back into the expression. We get \(\frac{16a^{20}b^{12}}{-16a^{20}b^7}\).
03
Simplify the Coefficients
The coefficients in the fraction are \(16\) and \(-16\). Divide these coefficients \(\frac{16}{-16} = -1\). Include this in the simplified expression: \(-1\cdot \frac{a^{20}b^{12}}{a^{20}b^7}\).
04
Simplify Using the Quotient of Powers Rule for a
For \(a^{20} \), use the quotient rule \((\frac{x^m}{x^n} = x^{m-n})\). Since the powers of \(a\) in the numerator and denominator are the same, \(a^{20-20} = a^0 = 1\).
05
Simplify Using the Quotient of Powers Rule for b
Now, apply the quotient rule to \(b\): \(b^{12-7} = b^5\).
06
Write the Final Simplified Expression
Combine the results from the previous steps. The simplified expression is: \(-1 \cdot 1 \cdot b^5 = -b^5\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Power of a Power Rule
The "Power of a Power" rule is a fundamental principle in algebra that assists in simplifying expressions where the same base is raised to multiple exponents. In simpler terms, when you have \[ (x^m)^n = x^{m\cdot n} \]this means that you multiply the exponents together. This rule is particularly useful when dealing with expressions like \[ (2a^5b^3)^4 \]. Here's how it works:
- Firstly, each term inside the parentheses is raised to the power of 4.
- For numbers, like 2, this is straightforward: \[ 2^4 = 16 \].
- For variables, multiply their exponents: \[ (a^5)^4 = a^{5\cdot4} = a^{20} \] and \[ (b^3)^4 = b^{3\cdot4} = b^{12} \].
Quotient of Powers Rule
The "Quotient of Powers" rule allows you to simplify expressions that involve dividing exponential terms with the same base. According to this rule, \[ \frac{x^m}{x^n} = x^{m-n} \]. Essentially, you subtract the exponent in the denominator from the exponent in the numerator. This rule is very helpful when simplifying expressions such as \[ \frac{16a^{20}b^{12}}{-16a^{20}b^7} \].
- Firstly, evaluate the coefficients \[ \frac{16}{-16} = -1 \]. Coefficients are ordinary numbers and are treated separately from variables.
- For the variable \[ a \], since \[ a^{20} \] is the same in both the numerator and denominator, it simplifies to \[ a^{20-20} = a^0 \]. Since any number to the power of zero is 1, the \[ a \] terms cancel out.
- For the variable \[ b \], apply the rule: \[ b^{12-7} = b^5 \].
Simplifying Expressions
Simplifying expressions involves breaking down complex algebraic expressions into their simplest form. This usually entails deploying various algebraic rules, such as the power of a power and the quotient of powers. Let's explore how to effectively simplify expressions like \[ \frac{(2a^5b^3)^4}{-16a^{20}b^7} \].
- Start by simplifying each part of the expression individually. Use the power of a power rule to handle terms inside parentheses.
- Simplify the numerator first to make calculations easier later on.
- Substitute back the simplified form into the original expression.
- Simplify the fraction by dealing with coefficients separately. For instance, \[ \frac{16}{-16} = -1 \].
- Next, handle the variable terms using the quotient of powers rule.
