Problem 26 - College Algebra

Question

Simplify each exponential expression $$ x^{7} y^{0} $$

Step-by-Step Solution

Answer

The simplified form of the expression \( x^{7} y^{0} \) is \( x^{7} \).

Step 1: Identification

Identify the parts of the expression. The expression given is \( x^{7} y^{0} \). This expression is a product of two terms: \( x^{7} \) and \( y^{0} \).

Step 2: Application of the Rule

Recall the exponent law that anything to the power of zero is equal to one, excluding zero itself. Apply this law to the second term of the expression, \( y^{0} \). So, \( y^{0} = 1 \).

Step 3: Simplification

Substitute \( y^{0} = 1 \) into the original expression, we get \( x^{7} * 1 \). This simplifies to \( x^{7} \).

Key Concepts

Exponent Rules

When dealing with expressions involving exponents, it is essential to understand and apply the correct exponent rules. These rules help us to simplify expressions and solve various mathematical problems easily. Here are some of the basic exponent rules:

• Product of Powers Rule: When multiplying two expressions with the same base, we add their exponents together. For example, \( a^m \cdot a^n = a^{m+n} \).
• Power of a Power Rule: To simplify an expression raised to another exponent, multiply the exponents. That means \((a^m)^n = a^{mn} \).
• Power of a Product Rule: When raising a product to a power, distribute the exponent to each factor in the product: \((ab)^m = a^m b^m \).
• Quotient of Powers Rule: When dividing two expressions with the same base, subtract the exponent of the denominator from that of the numerator: \( \frac{a^m}{a^n} = a^{m-n} \).
• Zero Exponent Rule: Any non-zero number raised to the zero power is 1: \( a^0 = 1 \) where \( a eq 0 \).

Understanding these rules helps us solve and simplify exponential expressions efficiently.

Zero Exponent

One of the fundamental exponent rules is the zero exponent rule. This rule states that any non-zero number raised to the power of zero equals one. This might seem a bit counterintuitive at first, but it has a logical basis and simplifies many expressions.To understand why this rule makes sense, consider the pattern in the powers of a number. Take for example the sequence of powers of 2:

• \( 2^3 = 8 \)
• \( 2^2 = 4 \)
• \( 2^1 = 2 \)
• If we continue the pattern, dividing each term by 2, we reach \( 2^0 = 1 \).

The zero exponent rule is crucial when simplifying expressions like \( x^7 y^0 \). Knowing \( y^0 = 1 \), the entire term becomes \( x^7 \cdot 1 = x^7 \). Thus, recognizing and applying the zero exponent rule reduces complexity in algebraic expressions.

Simplification of Expressions

Simplifying expressions involving exponents can make them much easier to understand and work with. The goal of simplification is to condense the expression as much as possible, using known rules, so it retains the same value but is more concise and manageable.Here are some tips for simplifying exponential expressions:

• Apply Exponent Rules: Use the exponent rules mentioned earlier to combine and reduce terms within the expression.
• Look for Zero Exponents: Identify any terms raised to the zero power and replace them with one, thereby simplifying the expression.
• Combine Like Terms: If the expression includes multiple terms with the same base, use appropriate rules to combine them.

For instance, in \( x^7 y^0 \), we simplify the expression by recognizing that \( y^0 = 1 \). This transforms the initial expression into \( x^7 \cdot 1 \), which is simply \( x^7 \). By practicing these steps, students can become proficient in simplifying exponential expressions.