Problem 53 - Algebra 1: Concepts and Skills

Question

Rewrite the expression with positive exponents. $$ x^{-2} y^{4} $$

Step-by-Step Solution

Answer

The expression \( x^{-2} y^{4} \) rewritten with positive exponents is \( \frac{1}{x^2} y^{4} \).

Step 1: Identify Negative Exponents

Identify any terms in your expression with negative exponents. In this case, the term with a negative exponent is \( x^{-2} \).

Step 2: Apply Negative Exponent Rule

Use the rule \( x^{-n} = \frac{1}{x^n} \) to rewrite the term with a negative exponent as a term with a positive exponent. Applying this rule gives us \( \frac{1}{x^2} \) for \( x^{-2} \).

Step 3: Write Final Expression with Positive Exponents

Combine the transformed term from Step 2 with other terms from the original expression to form the final expression with positive exponents. In this fall, we have \( \frac{1}{x^2} y^{4} \).

Key Concepts

Understanding Negative Exponents

Grasping the idea of negative exponents is crucial for mastering algebraic expressions. A negative exponent tells us how many times to divide by the number, rather than multiplying. It's like a direction to take the reciprocal of the base raised to the absolute value of the exponent.

For instance, the expression with a negative exponent, such as \( x^{-2} \), can be interpreted as \( 1/x^2 \) instead of multiplying \( x \) by itself twice. This shift from multiplication to division is the key to transforming negative exponents into positive ones.

Here's a tip: remember that a negative exponent doesn't make the number negative; it just moves it to the other side of the fraction line. If the number was in the numerator, it will go to the denominator, and vice versa.

It's also useful to know that any number, except for zero, raised to the power of zero is always one. This helps in understanding the continuity of the exponent rules.

Exponent Rules Demystified

Working with exponents requires understanding several key rules. These not only make it easier to simplify expressions but also to solve complex equations.

• Product Rule: When multiplying two exponents with the same base, you add the exponents: \( x^m \times x^n = x^{m+n} \).
• Quotient Rule: When dividing two exponents with the same base, you subtract the exponents: \( x^m \/ x^n = x^{m-n} \).
• Power Rule: When raising an exponent to another power, you multiply the exponents: \( (x^m)^n = x^{m \times n} \).
• Zero Exponent Rule: Any base (except zero) raised to the power of zero is one: \( x^0 = 1 \).
• Negative Exponent Rule: A negative exponent indicates reciprocal: \( x^{-n} = 1 \/ x^n \).

When you encounter a negative exponent, apply these exponent rules to rewrite the expression in positive exponent form. This helps in simplifying algebraic expressions and solving them more accurately.

Simplifying Algebraic Expressions

An algebraic expression may look complex, but it's merely a combination of numbers, variables, and arithmetic operations. The process of simplifying these expressions often involves applying exponent rules, combining like terms, and factoring.

For example, if faced with the expression \( x^{-2} y^{4} \), you should recognize that simplification will result in an equivalent expression with only positive exponents. By rewriting the term with the negative exponent according to the negative exponent rule, the expression becomes \( \frac{1}{x^2} y^{4} \).

Remember, combining like terms can further simplify the expression. If you had a similar term with \( y^{4} \) but with different coefficients, you would combine them to form a single term, thus making the expression more concise.