Problem 41 - Elementary Algebra

Question

Write \(\frac{x^{10} y^{3}(x+7)^{4}}{x^{-2} y^{3}(x+7)^{-1}}\) so that only positive exponents appear.

Step-by-Step Solution

Answer

Question: Rewrite the expression \(\frac{x^{10} y^{3}(x+7)^{4}}{x^{-2} y^{3}(x+7)^{-1}}\) with only positive exponents. Answer: \(x^{12}(x+7)^{5}\)

Step 1: Identify the bases and exponents

In the given expression, we have the following bases and exponents: - Base: \(x\), exponents: \(10, -2\) - Base: \(y\), exponents: \(3, 3\) - Base: \((x+7)\), exponents: \(4, -1\)

Step 2: Apply the properties of exponents

Using the properties of exponents mentioned in the analysis section, we can rewrite the expression as: \(\frac{x^{10} y^{3}(x+7)^{4}}{x^{-2} y^{3}(x+7)^{-1}} = x^{10-(-2)} y^{3-3} (x+7)^{4-(-1)}\)

Step 3: Simplify the expression

Now we simply need to simplify the expression by performing the operations: \(x^{10-(-2)} y^{3-3} (x+7)^{4-(-1)} = x^{12} y^{0} (x+7)^{5}\) Since any base raised to the power of 0 is equal to 1, we have: \(x^{12} y^{0} (x+7)^{5} = x^{12} \cdot 1 \cdot (x+7)^{5} = x^{12}(x+7)^{5}\)

Step 4: Final Answer

The simplified expression with only positive exponents is: \(x^{12}(x+7)^{5}\)

Key Concepts

Exponents

Exponents are shorthand for repeated multiplication of the same thing by itself. For example, the exponential expression \( x^4 \) means \( x \) is multiplied by itself four times: \( x \times x \times x \times x \). Understanding exponents is critical for simplifying algebraic expressions, particularly when variables are raised to different powers.

When working with exponents, it's important to know the basic terminology. The number \( x \) is called the base, and the number \( 4 \) is called the exponent or power, which tells us how many times to multiply the base by itself. This concept extends to any real number as the base and any integer as the exponent, including negative and positive ones, which play a crucial role in simplification.

Properties of Exponents

Properties of exponents allow us to manipulate expressions involving powers in an algebraic manner. These properties are rules that tell us how to handle exponents when we multiply, divide, or raise them to a power. Here are a few key properties used in simplifying algebraic expressions:

• The Product of Powers Rule: \( a^m \times a^n = a^{m+n} \), which tells us that when multiplying two powers that have the same base, you can add the exponents.
• The Quotient of Powers Rule: \( a^m \div a^n = a^{m-n} \), stating that when dividing two powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator.
• The Power of a Power Rule: \( (a^m)^n = a^{m \times n} \), which tells us to multiply the exponents when raising a power to another power.
• The Zero Exponent Rule: \( a^0 = 1 \), for any nonzero base \( a \), the expression with an exponent of zero is always equal to one.

Applying these properties helps in simplifying expressions, especially when different operations involving exponents are involved.

Algebraic Manipulation

Algebraic manipulation is the process of applying mathematical operations to simplify or rearrange algebraic expressions. It taps into a variety of rules and properties, including the properties of exponents, distributive property, and combining like terms. When faced with an expression like \( \frac{x^{10} y^{3}(x+7)^{4}}{x^{-2} y^{3}(x+7)^{-1}} \), algebraic manipulation encompasses identifying like bases and applying exponent rules to combine or eliminate terms.

An important aspect is recognizing that any term with an exponent of zero, such as \( y^{0} \), becomes one, drastically simplifying the expression. This type of manipulation requires careful observation and methodical application of rules to ensure that each step logically follows from the previous one, eventually arriving at a simplified form. In this exercise, algebraic manipulation transforms the original complex fraction into a much simpler expression \( x^{12}(x+7)^{5} \) with only positive exponents.