Problem 121 - Algebra and Trigonometry Real Mathematics, Real People

Question

Simplify the expression. $$\left(5 s^{5} t^{-5}\right)\left(\frac{3 s^{-2}}{50 t^{-1}}\right)$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(0.3s^3t^{-6}\).

Step 1: Simplify the numerical coefficients

First, deal with the numerical components of the expression, which involve multiplication and division: \(5 \cdot \frac{3}{50} = \frac{15}{50} = 0.3\).

Step 2: Simplify variable of \(s\)

Moving on to the variable \(s\), apply the rules of exponents for multiplication and division. This requires summing the exponent when the bases (the variable) are the same: \(s^{5} \cdot s^{-2} = s^{(5-2)} = s^3\).

Step 3: Simplify variable of \(t\)

Lastly, for the variable \(t\), apply the same rules of exponents to get: \(t^{-5} \cdot t^{-1} = t^{(-5-1)} = t^{-6}\).

Step 4: Combine simplified components

Now combine the simplified numerical coefficient and the simplified variables to give the final simplified expression: \(0.3 \cdot s^3 \cdot t^{-6}\).

Key Concepts

Exponents and Powers

Exponents and powers are essential components of algebra that express repeated multiplication of the same number or variable. An exponent is a small number placed to the upper right of a base number or variable. It indicates how many times the base is used as a factor. For example, in the expression \(s^5\), the base \(s\) is multiplied by itself 5 times.

When simplifying expressions with exponents, it’s crucial to understand how to handle negative exponents and fractional exponents:

• A negative exponent such as \(t^{-1}\) indicates a reciprocal: \(1/t\).
• A power of zero means that the base equals one: \(n^0 = 1\), regardless of \(n\) (with the exception of \(n=0\)).

Understanding these concepts will help you simplify algebraic expressions efficiently and accurately.

Rational Expressions

Rational expressions are fractions that have polynomials in their numerator and denominator. Simplifying these expressions requires understanding how to factor polynomials effectively and how to cancel out common factors:

• Ensure both the numerator and the denominator are completely factored.
• After factoring, identify and cancel common factors to simplify the expression.

For example, when dealing with coefficients and variables like \(\frac{3 s^{-2}}{50 t^{-1}}\), you treat the numerical part \(\frac{3}{50}\) just like a regular fraction. Simplify it by dividing the greatest common divisor, just as you would with any fractional coefficient.

Understanding how to simplify rational expressions is crucial in algebra as it makes solving and understanding equations easier, particularly when these equations become part of larger algebraic problems and word problems.

Rules of Exponents

The rules of exponents are mathematical guidelines used to simplify expressions involving powers and exponents. Knowing how to apply these rules correctly helps you simplify complex algebraic expressions by reducing the number of terms and factors. Here are some important exponent rules:

• Product of Powers Rule: When multiplying two expressions with the same base, add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• Quotient of Powers Rule: When dividing two expressions with the same base, subtract the exponents: \(a^m / a^n = a^{m-n}\).
• Power of a Power Rule: Take an exponent to another power by multiplying the exponents: \((a^m)^n = a^{m\cdot n}\).
• Negative Exponent Rule: An expression with a negative exponent represents the reciprocal with a positive exponent: \(a^{-m} = \frac{1}{a^m}\).

By using these exponent rules, you can transform complex expressions into simpler and more manageable forms, which is especially helpful in solving equations and inequalities in algebra.