Problem 24 - Algebra 2 and Trigonometry

Question

Simplify each expression. In each exercise, all variables are positive. \(\left(\frac{2 a^{3}}{a^{2}}\right)^{5} \cdot b\)

Step-by-Step Solution

Answer

The simplified expression is \( 32a^5b \).

Step 1: Identify the Inner Fraction

Start by simplifying the inner fraction \( \frac{2a^3}{a^2} \). Divide the terms of the numerator and the denominator. Since \( a^3 \) divided by \( a^2 \) is \( a^{3-2} \), it simplifies to \( a^1 \). Therefore, \( \frac{2a^3}{a^2} = 2a \).

Step 2: Apply the Power Rule

Take the simplified expression \( 2a \) from the previous step and apply the outer power: \( (2a)^5 \). According to the power rule, \( (ab)^n = a^n b^n \). Apply this to get \( 2^5 a^5 \). So \( (2a)^5 = 32a^5 \).

Step 3: Combine with Other Terms

Now combine \( 32a^5 \) with the term \( b \). This does not require any further simplification, resulting in \( 32a^5b \).

Key Concepts

Simplification

Simplifying algebraic expressions is all about making them easier to work with or understand. It usually involves reducing expressions to their most basic forms.
Variables and constants are simplified by combining like terms or eliminating excess operations.
For example, consider the expression \( \frac{2a^3}{a^2} \). In this case, the goal is to simplify the fraction by reducing it to its simplest form.

• We can do this by canceling one \(a\) term from both the numerator and the denominator.
• This reduces \(a^3\) divided by \(a^2\) to \(a^{3-2}\) or simply \(a^1\).

Thus, the original fraction simplifies down to \(2a\). This process is important because it minimizes any unnecessary complexity, making the problem more straightforward to solve.

Power Rule

The Power Rule is a fundamental principle in mathematics when dealing with exponents.
It tells us how to handle expressions where both multiplication and powers are included.
In algebra, the rule is stated as \((ab)^n = a^n b^n\), meaning you apply the power to each component separately.

• For example, if you have \((2a)^5\), you apply the power of 5 to both 2 and \(a\) individually.
• First, calculate \(2^5\), which results in 32, and then \(a^5\), maintaining the variable's power.

So \((2a)^5\) becomes \(32a^5\). By applying the Power Rule, we keep our solutions precise and effectively manage more complex equations involving exponents.

Fraction Division

Fraction division simplifies ratios of algebraic expressions. It involves breaking down complex numerators and denominators.
This requires understanding how terms across a division sign are related.
Take, for example, the fraction \(\frac{2a^3}{a^2}\).

• To divide this, recognize that the division translates to subtracting exponents: \(a^3 / a^2 = a^{3-2}\).
• After canceling like terms and simplifying, it leaves us with \(2a\).

Learning to efficiently divide fractions helps in tackling problems with multiple variable terms, keeping the expression efficient and tidy.

Exponents

The concept of exponents deals with repeated multiplication of a base number.
Exponents are written as superscript numbers, indicating how many times to multiply the base by itself.
For instance, in the expression \(a^3\), the exponent "3" signifies that \(a\) is multiplied by itself three times.

• When exponents appear in fractions, like \(\frac{a^3}{a^2}\), subtract the denominator's exponent from the numerator's.
• This gives \(a^{3-2} = a\).

Understanding exponent rules and operations is vital for algebraic manipulations and problem-solving, as it streamlines expressions and calculations by condensing repetitive multiplications into compact forms.