Question

Evaluate. $$4\left(\frac{a^{7}}{y^{4}}\right)^{0}$$

Short answer

\(4 \times 1 = 4\). The expression evaluates to 4.

Step-by-step solution

Apply the Zero Exponent Rule

Any non-zero number raised to the power of zero is equal to 1. Therefore, regardless of the values of \(a\) and \(y\) (as long as they are not zero), the expression inside the parentheses raised to the zero power will be equal to 1.

Calculate the Final Answer

After applying the zero exponent rule, the expression simplifies to \(4 \times 1\). Multiply 4 by 1 to obtain the final answer.

Key concepts

Exponents in Mathematics

Exponents, also known as powers or indices, are a fundamental part of algebra that denote the repeated multiplication of a base number. For instance, when we see an expression like \( 3^4 \), this implies that the base number 3 is multiplied by itself 4 times: \( 3 \times 3 \times 3 \times 3 \). This can be a daunting concept for some students, especially as they encounter more complex expressions involving variables, such as \( a^7 \) or \( y^4 \).

Understanding exponents is crucial not only for simplifying algebraic expressions but also for grasping higher-level math topics. When we look at the expression from the exercise, \(4 \left(\frac{a^{7}}{y^{4}}\right)^{0}\), it allows us to apply a very important exponent rule that can dramatically simplify our calculations—the zero exponent rule.

It's essential to remember that exponents are not just about dealing with numbers; they also play a significant role in scientific notation, growth and decay models, and even in financial calculations involving interest rates.

Simplifying Algebraic Expressions

In simplifying algebraic expressions, one aims to reduce the complexity of the expression, making it easier to understand or solve. This often involves combining like terms, factoring, expanding, and applying exponent rules, among others. A crucial part of this process is being methodical and understanding the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

In algebra, simplicity is key. For example, when presented with the expression in our exercise, \(4 \left(\frac{a^{7}}{y^{4}}\right)^{0}\), you might initially think there's a lot of work ahead. However, the zero exponent rule offers a wonderful shortcut, allowing us to transform any expression with an exponent of zero into the number one—essentially neutralizing what could be a complex calculation. This exemplifies the power of understanding and applying rules of algebra to simplify expressions efficiently and correctly.

Properties of Exponents

Properties of exponents are rules that describe how to handle expressions with exponents during operations such as multiplication, division, and exponentiation. Some of the key properties include the product rule \((x^a \cdot x^b = x^{a+b})\), the quotient rule \((\frac{x^a}{x^b} = x^{a-b} )\), the power of a power rule \(( (x^a)^b = x^{a \cdot b} )\), and the one under discussion— the zero exponent rule \((x^0 = 1)\), where 'x' is any non-zero number.

These rules are indispensable tools when dealing with algebraic expressions. They provide a structured approach that enables students to tackle seemingly complex expressions systematically. When applied correctly, as in Step 1 of our solved exercise, they can take an expression that looks intimidating and reduce it to its simplest form with ease and precision, demonstrating their practical significance in mathematics.