Problem 34 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ x^{7} y^{-5} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression $$x^{7}y^{-5}$$ with positive exponents only. Answer: $$x^{7}y^{-5} = x^{7} \cdot \frac{1}{y^{5}}$$

Step 1: Identify the negative exponent

In our expression $$x^{7}y^{-5}$$, the exponent of "y" is negative (-5).

Step 2: Apply the rule for negative exponents

By using the rule for negative exponents, we will take the reciprocal of y and change the exponent from -5 to 5. $$ x^{7}y^{-5} = x^{7} \cdot \frac{1}{y^{5}} $$

Step 3: Write the final expression with positive exponents

Now we have the given expression with positive exponents only: $$ x^{7}y^{-5} = x^{7} \cdot \frac{1}{y^{5}} $$

Key Concepts

Exponent Rules

Understanding exponent rules is fundamental when dealing with algebraic expressions. An exponent tells us how many times to multiply a base number by itself. For example, in the expression \( x^3 \), the base is \( x \) and the exponent is 3, which means \( x \) is multiplied by itself three times: \( x \times x \times x \).

There are a few key rules to remember:

• The product rule: \( x^a \times x^b = x^{a+b} \)
• The quotient rule: \( \frac{x^a}{x^b} = x^{a-b} \)
• The power of a power rule: \( (x^a)^b = x^{a\times b} \)
• The rule for negative exponents: \( x^{-a} = \frac{1}{x^a} \)

Using the rule for negative exponents, we can convert negative exponents into positive ones by taking the reciprocal of the base and making the exponent positive, as seen in the exercise with \( x^{7}y^{-5} = x^{7} \cdot \frac{1}{y^{5}} \). This process ensures we only have positive exponents in the final expression.

Positive Exponents

Positive exponents represent the number of times a base number is multiplied by itself. They are straightforward to interpret and calculate. For example, \( 2^4 \) means 2 is multiplied by itself 4 times, giving us 16.

When dealing with positive exponents, remember that:

• Any non-zero number raised to the power of 0 is 1: \( x^0 = 1 \).
• \( x^1 = x \), meaning the base is unchanged when raised to the power of 1.

These rules help simplify expressions and solve algebraic equations. In the exercise, after adjusting for negative exponents, we are left with \( x^7 \), which indicates that \( x \) would be multiplied by itself 7 times.

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and operations such as addition, subtraction, multiplication, and division, including exponents. They represent mathematical relationships and can be simple or complex.

The expression \( x^{7}y^{-5} \) is an algebraic expression containing two variables, \( x \) and \( y \), with respective exponents. To simplify and manipulate these expressions, we apply exponent rules. It's important to remember that when simplifying expressions, each part of the expression needs attention—constants, coefficients (numbers in front of variables), variables, and exponents.

Exercise Improvement Advice

In improving exercises like \( x^{7}y^{-5} \), it's beneficial to:

• Explain each step in detail and why it's necessary
• Clarify any misconceptions about negative exponents
• Provide multiple examples to practice correct application
• Highlight the importance of keeping variables nonzero (since \( x^0 \) is undefined)

This approach can help students grasp the concept more thoroughly and apply it confidently in various situations.