Question

Write each expression without parentheses. Assume all variables are positive. $$ \left(\frac{c^{3}}{d}\right)^{4} $$

Short answer

Answer: The simplified form of the expression is $\frac{c^{12}}{d^{4}}$.

Step-by-step solution

Apply the power-to-power rule

Since the entire expression inside the parentheses has an exponent of 4, we apply the power-to-power rule. This rule states that \((a^{m})^{n} = a^{mn}\), so we multiply the exponent of each term inside the parentheses by 4: $$ \left(\frac{c^{3}}{d}\right)^{4} = \frac{(c^{3})^{4}}{(d)^{4}} $$

Simplify the terms

Now, simplify each term by carrying out the exponents' multiplication: $$ \frac{(c^{3})^{4}}{(d)^{4}} = \frac{c^{12}}{d^{4}} $$

Write the final answer

We have simplified the expression without parentheses. The final answer is: $$ \frac{c^{12}}{d^{4}} $$

Key concepts

Exponents

Exponents are a fundamental aspect of algebra that express how many times a number, known as the base, is multiplied by itself. For example, in the expression \(c^3\), \(c\) is the base and the exponent is 3, indicating \(c\) is multiplied by itself three times: \(c \cdot c \cdot c\).

Understanding exponents is critical for simplifying expressions and solving equations. They are a shorthand method that can simplify the representation and calculation of large numbers or repetitive multiplication.

• Positive exponents denote repeated multiplication.
• Negative exponents represent the reciprocal or division.
• A zero exponent means the quantity is equal to one (as long as the base isn't zero).

Exponents are used in numerous algebra rules, including the power rule, which we'll discuss next.

Power Rule

The power rule is a handy guide in managing expressions with exponents. It states that when you raise a power to another power, you multiply the exponents.

For instance, consider the expression \((a^m)^n\). Instead of multiplying \(a^m\) by itself \(n\) times, you can simply multiply the exponents \(m\) and \(n\) to get \(a^{mn}\). This rule streamlines computations and clarifies expressions.

Application of Power Rule

When given a term like \((\frac{c^3}{d})^4\), apply the power rule separately to the numerator and the denominator:

• Multiply 3 (from \(c^3\)) by 4 to get \(c^{12}\).
• Multiply the implicit exponent 1 of \(d\) by 4 to achieve \(d^4\).

This results in the simplified expression \(\frac{c^{12}}{d^{4}}\).

Simplifying Expressions

Simplifying expressions involves reducing them to their most elementary form to make them easier to work with or understand. This process often utilizes rules such as those pertaining to exponents.

In algebra, simplification can include eliminating parentheses, consolidating terms, or reducing fractions to their simplest form. Let's consider the expression \((\frac{c^3}{d})^4\):

• Use the power rule to expand exponents: from \((c^3)^4\) to \(c^{12}\) and \(d^4\).
• Rewrite the expression without parentheses: \(\frac{c^{12}}{d^{4}}\).

This process clarifies the expression and makes it more manageable, aiding further mathematical operations or evaluations.