Problem 123 - Elementary Algebra

Question

For the following problems, write each expression so that only positive exponents appear. $$ \left(x^{7}\right)^{-4} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(\left(x^{7}\right)^{-4}\) with a positive exponent. Answer: \(\frac{1}{x^{28}}\)

Step 1: Recall the property of exponents for powers raised to powers

When raising a power to another power, we can simplify the expression by multiplying the exponents. The property is \((a^m)^n = a^{mn}\). In this problem, we have: $$ \left(x^{7}\right)^{-4} $$

Step 2: Apply the property of exponents to our expression

We can now apply the property of exponents for powers raised to powers to our expression, which means multiplying the exponents: $$ \left(x^{7}\right)^{-4} = x^{7 \times (-4)} $$

Step 3: Multiply the exponents

Now, we will multiply the exponents: $$ x^{7 \times (-4)} = x^{-28} $$

Step 4: Use the property of exponents to rewrite with positive exponents

To write our expression with a positive exponent, we can use the reciprocal property of exponents: \(a^{-n} = \frac{1}{a^n}\). Applying this property to our expression, we get: $$ x^{-28} = \frac{1}{x^{28}} $$ So, our final answer with a positive exponent is: $$ \left(x^{7}\right)^{-4} = \frac{1}{x^{28}} $$

Key Concepts

Positive Exponents

Exponents are a way of expressing repeated multiplication of a number by itself. When you see a positive exponent, it tells you how many times to multiply the base number. For example, if you have \( x^3 \), it means \( x \times x \times x \). Positive exponents are straightforward because they just tell you to keep multiplying.
When dealing with expressions, it is often easier to work with positive exponents. They avoid the complexity of fractions that can arise with negative exponents.
In mathematical problems, expressing your final answer with positive exponents is preferred, as it simplifies the expression and makes it easier to understand and compare.

Properties of Exponents

The properties of exponents are key tools for simplifying expressions. Understanding these properties makes working with exponential expressions much easier.

• Product of Powers: When multiplying similar bases, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• Power of a Power: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{mn} \).
• Power of a Product: Distribute the exponent to each factor: \( (ab)^n = a^n \times b^n \).
• Zero Exponent: Any number raised to the power of zero is 1: \( a^0 = 1 \), where \( a eq 0 \).

Using these properties can help in transforming expressions to a more manageable form. In our specific problem, we used the "Power of a Power" property to multiply the exponents and simplify the expression.

Reciprocal Property

The reciprocal property is very useful when dealing with negative exponents. A negative exponent means you take the reciprocal of the base raised to the corresponding positive exponent.
For example, \( a^{-n} \) is equivalent to \( \frac{1}{a^n} \). This property helps in converting negative exponents into positive ones, which are easier to interpret and work with.
In our solution, the reciprocal property allowed us to transform \( x^{-28} \) into \( \frac{1}{x^{28}} \), resulting in an expression that contains only positive exponents.
Remember, the reciprocal property simply adds a 'flip', moving the base from the numerator to the denominator and changing the exponent to a positive value.