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Exponents are used to represent repeated multiplication of a number by itself. For instance, instead of writing \( 2 \times 2 \times 2 \times 2 \), we can write \( 2^4 \). Understanding how to manipulate exponents is crucial in algebra. Key rules include:

• When multiplying terms with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• When dividing terms with the same base, you subtract the exponents: \( a^m \div \ a^n = a^{m-n} \).
• Any number raised to the power of zero is 1: \( a^0 = 1 \).

Mastering these rules helps in simplifying expressions efficiently.

Fractions represent a part of a whole and are expressed as \frac{a}{b}\, where \( a \) is the numerator and \( b \) is the denominator. In algebra, fractions often combine with exponents, and understanding how to simplify them is important. Key things to remember are:

• Combining fractions by finding a common denominator.
• Simplifying fractions by reducing the numerator and the denominator, dividing both by their greatest common divisor (GCD).
• When dealing with exponents in fractions, treat the numerator and the denominator separately, applying exponent rules to each part.

Let’s apply these to our exercise: combine the numerators first, then simplify the resulting fraction.

Breaking problems into clear, manageable steps helps in understanding and solving complex algebraic expressions. Here’s a breakdown of the given exercise using step-by-step solutions:

Step 1: Combine the Numerators
In the numerator, we have \( p^2 \times p^9 \). According to the exponent multiplication rule, we add the exponents: \( p^{2 + 9} = p^{11} \).

Step 2: Simplify the Fraction
Now, we simplify the fraction \frac{p^{11}}{p^4}\. Using the division rule for exponents, subtract the exponent in the denominator from the numerator: \( p^{11 - 4} = p^7 \).

Step 3: Verify the Exponents
Finally, check that the exponents in your solution are positive. In this case, the exponent is 7, which is positive, so the final simplified expression is \( p^7 \). Breaking down problems like this ensures accuracy and boosts problem-solving confidence.

When simplifying expressions, it's often required to express the answer with positive exponents. Positive exponents indicate how many times to use the base as a multiplier. Negative exponents, on the other hand, represent reciprocal operations. For instance, \( a^{-n} = \frac{1}{a^n} \). In the given exercise, the final step confirmed the result has positive exponents:

• Starting with \( \frac{p^2 \times p^9}{p^4} \), we combined the exponents in the numerator: \( p^{2 + 9} = p^{11} \).
• Next, we simplified the fraction to ensure the exponent remains positive: \( p^{11-4} = p^7 \).

Writing answers with positive exponents is a standard requirement unless the context of the problem specifies otherwise. Always double-check to ensure your final expression adheres to this rule.