Problem 80
Question
\(\left(\frac{3 a}{5 b}\right)^{2}(2 a)^{2}\)
Step-by-Step Solution
Verified Answer
\frac{36a^4}{25b^2}
1Step 1: Simplify the First Fraction
Take the fraction \(\frac{3 a}{5 b}\) and square both the numerator and the denominator. This gives \(\frac{(3 a)^2}{(5 b)^2}\).
2Step 2: Apply the Powers
Square the numerator and denominator separately: \( (3 a)^2 = 9 a^2 \) and \( (5 b)^2 = 25 b^2\). Therefore, \(\frac{(3a)^2}{(5b)^2} = \frac{9a^2}{25b^2}\).
3Step 3: Simplify the Second Term
Square the second term \(2a\) to get \( (2a)^2\). This results in \( (2a)^2 = 4a^2 \).
4Step 4: Multiply the Simplified Terms
Multiply the simplified fraction \(\frac{9a^2}{25b^2}\) by the squared term \((4a^2)\). This gives \(\frac{9a^2}{25b^2} \times 4a^2 = \frac{9 \times 4 \times a^2 \times a^2}{25b^2} = \frac{36a^4}{25b^2}\).
Key Concepts
Simplifying FractionsExponentiationPolynomial MultiplicationRational Expressions
Simplifying Fractions
Simplifying fractions is essential in algebra to make expressions more manageable. A fraction represents a division of two quantities. Simplifying involves reducing the numerator (top part) and the denominator (bottom part) to their smallest values.
In our problem, we start with the fraction \(\frac{3a}{5b}\).
To simplify this, we need to apply the exponent to both the numerator and the denominator. Squaring them results in \(\frac{(3a)^2}{(5b)^2} = \frac{9a^2}{25b^2}\). Now, the fraction is simpler and easier to work with.
Remember, simplifying fractions helps reduce errors and makes subsequent computations more straightforward.
In our problem, we start with the fraction \(\frac{3a}{5b}\).
To simplify this, we need to apply the exponent to both the numerator and the denominator. Squaring them results in \(\frac{(3a)^2}{(5b)^2} = \frac{9a^2}{25b^2}\). Now, the fraction is simpler and easier to work with.
Remember, simplifying fractions helps reduce errors and makes subsequent computations more straightforward.
Exponentiation
Exponentiation involves raising a number or variable to a power. This means multiplying the number by itself as many times as the exponent indicates.
In our exercise, we see exponentiation in multiple places. For instance, \((3a)^2\) and \((5b)^2\).
When squaring \((3a)\), we do \(3a \times 3a = 9a^2\). Similarly, squaring \(5b\) gives \(25b^2\).
Understanding exponentiation is crucial because it helps in expanding and simplifying expressions effectively.
Always remember, \( (ab)^n = a^n \times b^n \) and \( (a^m)^ = a^{mn} \). Applying these rules makes handling powers in algebra simpler.
In our exercise, we see exponentiation in multiple places. For instance, \((3a)^2\) and \((5b)^2\).
When squaring \((3a)\), we do \(3a \times 3a = 9a^2\). Similarly, squaring \(5b\) gives \(25b^2\).
Understanding exponentiation is crucial because it helps in expanding and simplifying expressions effectively.
Always remember, \( (ab)^n = a^n \times b^n \) and \( (a^m)^ = a^{mn} \). Applying these rules makes handling powers in algebra simpler.
Polynomial Multiplication
Polynomial multiplication involves expanding expressions where terms are added together. Here, when multiplying two polynomials, each term of one polynomial is multiplied by each term of the other.
In our problem, after simplifying the fraction, we multiply \(\frac{9a^2}{25b^2}\) by \(4a^2\).
This results in \(\frac{9a^2 \times 4a^2}{25b^2} = \frac{36a^4}{25b^2}\).
When multiplying polynomials, always distribute each term carefully and combine like terms where possible. This ensures expressions remain accurate and simplified.
In our problem, after simplifying the fraction, we multiply \(\frac{9a^2}{25b^2}\) by \(4a^2\).
This results in \(\frac{9a^2 \times 4a^2}{25b^2} = \frac{36a^4}{25b^2}\).
When multiplying polynomials, always distribute each term carefully and combine like terms where possible. This ensures expressions remain accurate and simplified.
Rational Expressions
Rational expressions are fractions that have polynomials in both the numerator and denominator. They appear frequently in algebra and calculus.
To work with rational expressions effectively, you often need to simplify, factor, and sometimes find common denominators.
In this exercise, our final solution is \(\frac{36a^4}{25b^2}\), which is a rational expression.
Always check if the rational expression can be simplified by factoring out common terms or canceling them. Reducing rational expressions makes them much easier to handle and understand in higher-level math problems.
To work with rational expressions effectively, you often need to simplify, factor, and sometimes find common denominators.
In this exercise, our final solution is \(\frac{36a^4}{25b^2}\), which is a rational expression.
Always check if the rational expression can be simplified by factoring out common terms or canceling them. Reducing rational expressions makes them much easier to handle and understand in higher-level math problems.
Other exercises in this chapter
Problem 80
The length of one side of a triangle is \(a \mathrm{ft}\). The other sides of the triangle are \(6 \mathrm{ft}\) longer and \(9 \mathrm{ft}\) shorter than this
View solution Problem 80
\(7.9 \times 10^{4} \mathrm{~km}+6.8 \times 10^{5} \mathrm{~km}\)
View solution Problem 81
Problem: Use long division to simplify $$ \left(2 x^{2}+5 x+3\right) \div(x-8) $$ Incorrect Answer: $$ \begin{array}{r} x - 8 \longdiv { 2 x ^ { 2 } + 5 x + 3 }
View solution Problem 81
The height of a triangle is \(6 \mathrm{~cm}\) less than the base. a. If \(b=\) base, write a polynomial expression in \(b\) that represents the height, and dra
View solution