91Ó°ÊÓ

When working with exponents, the power of a power rule is a fundamental concept to understand. This rule helps you simplify expressions where an exponent is raised to another exponent. According to this rule, \[ (a^m)^n = a^{m \cdot n} \] which means you multiply the exponents together.
For instance, in the exercise, you have the expression \( \left(p^3\right)^{-2} \).Applying the rule means you multiply the exponents: \( 3 \times (-2) = -6 \),leading to \( p^{-6} \).
Similarly, applying the rule to the denominator \( 2^{-2} \)leaves \( 2^{-2} \) as it is, since it's simply \( 2^1 \) raised to \( -2 \).
This step simplifies the problem greatly and sets the stage for dealing with negative exponents.

Negative exponents often puzzle students, but they are easier to handle than they appear. A negative exponent indicates that the base should be moved to the opposite part of a fraction.
For a base \( a \)with a negative exponent \( -m \), the rule is\[ a^{-m} = \frac{1}{a^m} \].
This means that \( p^{-6} \)becomes \( \frac{1}{p^6} \),which moves \( p^6 \)to the denominator. For \( 2^{-2} \), it becomes \( \frac{1}{4} \),as \( 2^2 = 4 \).
The trick is simply remembering to "flip" the base with its positive exponent to the denominator or numerator to clear the negative sign. This crucial step turns complex expressions into manageable parts.

Simplifying fractions is often an essential final step in algebraic expressions, especially those involving exponents. When faced with a complex fraction such as\( \frac{1/p^6}{1/4} \),you simplify by multiplying by the reciprocal of the denominator.

• First, find the reciprocal of \( \frac{1}{4} \), which is \( 4 \).
• Next, multiply the fraction: \( \frac{1}{p^6} \cdot 4 \).
• This results in \( \frac{4}{p^6} \).

This simplification approach works efficiently, leading you to the final expression \( \frac{4}{p^6} \). Remember, multiplying by a reciprocal transforms the division into multiplication, cleaning up any complex fractions.