Textbook Question
Simplify each complex fraction. [ 1 + 1/(1-b) ] / [ 1 - 1/(1+b) ]
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0. Review of Algebra
Algebraic Expressions
3:45 minutesProblem 117
Textbook Question
Add or subtract as indicated. 12/25x^2+11/15x^3
Verified step by step guidance1
Step 1: Identify the terms in the expression. The given expression is \( \frac{12}{25}x^2 + \frac{11}{15}x^3 \). Notice that the terms have different powers of \(x\), so they are not like terms and cannot be directly added or subtracted.
Step 2: Factor out the greatest common factor (GCF) if possible. In this case, the coefficients \( \frac{12}{25} \) and \( \frac{11}{15} \) do not share a common factor, and the powers of \(x\) are different, so no factoring is possible.
Step 3: Rewrite the expression as it is already simplified. Since the terms are not like terms, the expression remains \( \frac{12}{25}x^2 + \frac{11}{15}x^3 \).
Step 4: If required, you can express the terms with a common denominator for the coefficients. The least common denominator (LCD) of \(25\) and \(15\) is \(75\). Rewrite each fraction with a denominator of \(75\): \( \frac{12}{25} = \frac{36}{75} \) and \( \frac{11}{15} = \frac{55}{75} \). The expression becomes \( \frac{36}{75}x^2 + \frac{55}{75}x^3 \).
Step 5: Combine the rewritten terms if necessary. Since the terms still have different powers of \(x\), they cannot be combined further. The final simplified expression is \( \frac{36}{75}x^2 + \frac{55}{75}x^3 \).
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Expressions
Rational expressions are fractions where the numerator and the denominator are polynomials. To add or subtract these expressions, it is essential to have a common denominator. This involves finding the least common multiple (LCM) of the denominators, which allows for the combination of the fractions into a single expression.
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Rationalizing Denominators
Finding a Common Denominator
Finding a common denominator is a crucial step in adding or subtracting rational expressions. The common denominator must be the least common multiple of the individual denominators. Once identified, each fraction is adjusted by multiplying its numerator and denominator by the necessary factors to achieve this common denominator, enabling the fractions to be combined.
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Combining Like Terms
Combining like terms involves simplifying expressions by adding or subtracting coefficients of terms that have the same variable raised to the same power. After obtaining a common denominator and combining the numerators, it is important to simplify the resulting expression by merging any like terms, which streamlines the final answer and makes it easier to interpret.
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