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Chapter 5: Problem 101

Simplify each expression, if possible. $$ \frac{s^{2} s^{2} s^{2}}{s^{3} s} $$

Short Answer

Expert verified
The simplified expression is \( s^2 \).

Step by step solution

01

Simplify the Numerator

In the numerator, we have repeated multiplications of the variable: \( s^2 \times s^2 \times s^2 \). To simplify this, recall that when you multiply powers with the same base, you can add the exponents: \( s^2 \times s^2 \times s^2 = s^{2+2+2} = s^6 \).
02

Simplify the Denominator

The denominator expression is \( s^3 \times s \). Using the rule of adding exponents for multiplication: \( s^3 \times s = s^{3+1} = s^4 \).
03

Divide Exponents

Now that we have \( \frac{s^6}{s^4} \), use the law of exponents that states when dividing like bases, subtract the exponent in the denominator from the exponent in the numerator: \( \frac{s^6}{s^4} = s^{6-4} = s^2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Laws of Exponents
The laws of exponents are a set of rules that simplify expressions involving powers of the same base. Understanding these rules makes it easier to handle algebraic expressions in math.
  • Product of Powers: When multiplying powers that have the same base, you need to add the exponents. For example, the expression \(a^m \times a^n\) simplifies to \(a^{m+n}\). This is because multiplying powers essentially means multiplying the base by itself multiple times.

  • Power of a Power: When raising a power to another power, you multiply the exponents. Thus, \((a^m)^n = a^{m \times n}\).

  • Quotient of Powers: When dividing powers with the same base, subtract the exponent in the denominator from the exponent in the numerator: \(\frac{a^m}{a^n} = a^{m-n}\).

  • Zero Exponent Rule: Any non-zero number raised to the power of zero equals 1: \(a^0 = 1\), as long as \(aeq0\).

  • Negative Exponent Rule: A negative exponent means that the base is on the wrong side of the division line, so \(a^{-n} \equiv \frac{1}{a^n}\).
Multiplying Powers
When working with algebraic expressions involving powers, it鈥檚 important to remember that multiplying powers with the same base is straightforward.In the expression \(s^2 \times s^2 \times s^2\), all terms share the base \(s\). To multiply them, simply add the exponents:
  • Step 1: Understand that each instance of \(s^2\) means \(s\) multiplied by itself.

  • Step 2: Apply the product of powers rule by adding the exponents together: \(2 + 2 + 2 = 6\).
  • Step 3: Write the simplified form: \(s^6\).
Using these steps helps you transform complex multiplication into a simpler expression, making it easier to work with in further computations.
Dividing Powers
Dividing powers that share the same base involves subtracting exponents. This is due to the idea of cancellation in division.Let鈥檚 consider the expression \(\frac{s^6}{s^4}\):
  • Step 1: Acknowledge that you are dealing with the same base in both the numerator and the denominator.

  • Step 2: Subtract the exponent of the denominator from the exponent of the numerator: \(6 - 4 = 2\).
  • Step 3: Express the result as \(s^2\).
This process shows how the power of the base reduces by the power in the divisor. Dividing powers efficiently converts complex divisions into more manageable terms, simplifying the problem overall.

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Most popular questions from this chapter

A special-product rule can be used to find \(31 \cdot 29\) $$ \begin{aligned} 31 \cdot 29 &=(30+1)(30-1) \\ &=30^{2}-1^{2} \\ &=900-1 \\ &=899 \end{aligned} $$ Use this method to find \(52 \cdot 48\).

Perform each division. $$ \frac{x^{3}-8}{x-2} $$

Communications. Telephone poles were installed every \((2 x-3)\) feet along a stretch of railroad track \(\left(8 x^{3}-6 x^{2}+5 x-21\right)\) feet long. What expression represents the number of poles that were used?

Perform the indicated operations. a. \((x y)^{2}\) b. \((x+y)^{2}\)

Perform each division. $$ \frac{3 b^{2}+11 b+6}{3 b+2} $$
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