Problem 37 - Algebra 1

Question

Rewrite the expression with positive exponents. $$\frac{1}{4 x^{-10} y^{14}}$$

Step-by-Step Solution

Answer

The expression \(\frac{1}{4x^{-10}y^{14}}\) when rewritten with positive exponents is \(\frac{y^{14}}{4x^{10}}\).

Step 1: Identify the term with a negative exponent

The term with a negative exponent in the fraction is \(x^{-10}\). The base here is x and its exponent is -10.

Step 2: Apply the rule of negative exponents

The rule of negative exponents states that any base with a negative exponent can be reciprocated (flipped) to make the exponent positive. Applying this rule, the term \(x^{-10}\) becomes \(x^{10}\) when transferred to the denominator.

Step 3: Rewrite the expression with positive exponents

By keeping the constant and positive exponent as is and by transferring the base with the negative exponent to the denominator to make it positive, the original expression \(\frac{1}{4 x^{-10} y^{14}}\) can be rewritten as \(\frac{y^{14}}{4 x^{10}}\).

Key Concepts

Positive Exponents

Exponents are shorthand for repeated multiplication of the same factor. When you have a positive exponent, it indicates how many times to multiply a number by itself. For instance, the expression

\(3^4\)

means you would multiply 3 by itself 4 times (3 * 3 * 3 * 3). The result of this operation is 81. Positive exponents streamline the process of working with large numbers and make it easier to perform operations and understand the scale of values in algebraic expressions.

Exponent Rules

Understanding exponent rules is critical in simplifying expressions and solving algebraic problems. The most common rules include:

• The Product Rule:
  
  \(a^m \cdot a^n = a^{m+n}\)
• The Quotient Rule:
  
  \(\frac{a^m}{a^n} = a^{m-n}\)
• The Power of a Power Rule:
  
  \((a^m)^n = a^{m \cdot n}\)
• The Negative Exponent Rule:
  
  \(a^{-n} = \frac{1}{a^n}\)
• The Zero Exponent Rule:
  
  \(a^0 = 1\), given that \(a eq 0\).

These rules allow you to manipulate and simplify expressions with confidence. For example, to handle negative exponents, you can use the Negative Exponent Rule by reciprocating the base and turning the exponent positive, which is a common step when simplifying expressions with negative exponents.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation signs. Variables are symbols (often letters) that stand in for unknown values. The beauty of algebraic expressions is that they can represent real-life situations and allow you to work with unknowns in a logical, systematic way.

For example, the expression

\(2x + 3y - 5\)

represents a combination of terms that are combined through addition and subtraction. When you're given values for \(x\) and \(y\), you can calculate the value of the entire expression. This feature of algebraic expressions makes them a fundamental aspect of algebra and vital for solving equations.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while maintaining their original value. It often requires combining like terms (terms that have the same variables to the same power), using exponent rules, and factoring when necessary.

For instance, if you have the expression

\(3x^2 + 4x - x^2 + 1\),

you'd combine like terms to simplify it to

\(2x^2 + 4x + 1\).

In the context of the original problem, simplifying expressions with negative exponents involves recognizing and applying the negative exponent rule, transforming those exponents into positive ones, and rewriting the expression in a straightforward, more digestible format. This ensures that the expressions are easy to interpret and work with on further algebraic operations.