Problem 44 - Algebra and Trigonometry

Question

Simplify the expression and eliminate any negative exponent(s). $$ \frac{\left(2 x^{3}\right)^{2}\left(3 x^{4}\right)}{\left(x^{3}\right)^{4}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{12}{x^2}\).

Step 1: Distribute the Exponents

The first task is to simplify by distributing the exponents across each part of the expression. This means applying the exponent to each base inside the parentheses. We have \((2x^3)^2\), which is equal to \(2^2 (x^3)^2\). Therefore, \((2x^3)^2 = 4x^6\) because \(2^2 = 4\) and \((x^3)^2 = x^{3 \times 2} = x^6\). Next, we have the expression \((3x^4)\) left as it is because it is to the power of 1.

Step 2: Simplify the Numerator

Within the numerator, after distributing the exponents, multiply the results: \(4x^6 imes 3x^4\). For the coefficients (numbers), \(4 \times 3 = 12\). For the exponents with the same base \(x\), add the exponents: \(x^6 imes x^4 = x^{6+4} = x^{10}\). Hence, the simplified numerator is \(12x^{10}\).

Step 3: Simplify the Denominator

Looking at the denominator \((x^3)^4\), apply the power rule: \((x^3)^4 = x^{3 \times 4} = x^{12}\). So the denominator simplifies to \(x^{12}\).

Step 4: Divide the Numerator by the Denominator

Now, simplify by dividing the numerator by the denominator. We have \(\frac{12x^{10}}{x^{12}}\). The rule for dividing exponents is to subtract the exponents: \(x^{10} \div x^{12} = x^{10-12} = x^{-2}\). Therefore, the expression simplifies to \(12x^{-2}\).

Step 5: Eliminate the Negative Exponent

To eliminate the negative exponent, rewrite \(x^{-2}\) as \(\frac{1}{x^2}\). The expression \(12x^{-2}\) becomes \(\frac{12}{x^2}\).

Key Concepts

Distributing Exponents

When simplifying expressions with exponents, it's crucial to understand how to distribute these exponents effectively. Distributing exponents involves applying the exponent on the outside of the parentheses to every element inside the parentheses. Take, for instance,

• The term \((2x^3)^2\).This means both 2 and \(x^3\) are raised to the power of 2.First, calculate \(2^2\), leading to 4.
  Next, apply this power to \(x^3\) by multiplying the exponents (3 and 2), resulting in \(x^{6}\).Therefore, \((2x^3)^2\) simplifies to \(4x^6\).Distributing exponents correctly is key to simplifying expressions accurately.

Negative Exponents

Negative exponents might look confusing at first, but remember, they indicate a reciprocal. For example, \(x^{-2}\) can be rewritten as \(1/x^{2}\).
To handle negative exponents in an expression, follow this simple rule: move the base with the negative exponent to the opposite part of the fraction.

• If it's in the numerator, move it to the denominator.
• If it's in the denominator, move it to the numerator.

This maneuver effectively turns the negative exponent into a positive one.
For instance, solving \(12x^{-2}\) requires converting \(x^{-2}\) to \(1/x^2\).So, the expression then becomes:\(\frac{12}{x^2}\).Negative exponents thus demand a switch in position within the fraction to achieve positive values.

Power Rule

The power rule in exponents is a straightforward yet powerful tool in algebra. It involves multiplying exponents when one power is raised to another. In our expression, we see this with the term \((x^3)^4\).

• Simply multiply the exponents:
• 3 and 4 result in 12.

Thus, \((x^3)^4\) simplifies to \(x^{12}\).This concept allows you to compress expressions significantly without solving each component multiple times separately. Consistent use of the power rule can streamline solving complex algebraic expressions.

Multiplying Exponents

When multiplying terms with the same base and exponents, adding the exponents is the rule to follow. This makes operations within expressions much simpler.In the expression part \(4x^6 \times 3x^4\), an essential step involves:

• Multiplying coefficients (the numbers) first: \(4 \times 3 = 12\).
• For the exponents with the same base 'x', add the exponents: \(x^{6} \times x^{4} = x^{6+4} = x^{10}\).

By multiplying the exponents, you simplify the expression to \(12x^{10}\) effectively. This method maintains accuracy while cutting down on cumbersome calculations, making it easier to address algebraic expressions efficiently.