Question

The expression \(\frac{a^{4}}{a^{6}}\) can be simplified by using the \(\underline{?}\) property.

Short answer

The property used to simplify the expression \(\frac{a^{4}}{a^{6}}\) is called 'Quotient of Powers Property'.

Step-by-step solution

Understand the properties of exponents

In the case of exponents, there are some properties that apply. One of the properties of exponents says that \(a^{n} / a^{m} = a^{n-m}\) where \(a\) is the base and \(n\) and \(m\) are the exponents.

Apply the property to the given expression

Applying this above mentioned property to the given expression \(\frac{a^{4}}{a^{6}}\) we have \(a^{4-6} = a^{-2}\). The property used here is the quotient of powers property.

Key concepts

Quotient of Powers Property

When you encounter a fraction with the same base raised to different exponents, such as \(\frac{a^m}{a^n}\), you can simplify the expression by subtracting the exponents. This rule is known as the quotient of powers property. Basically, when you're dividing like bases, you subtract the exponent of the denominator from the exponent of the numerator. Hence, \(\frac{a^m}{a^n} = a^{m-n}\). It's important to note that this property only applies when the base of the exponents - the 'a' in our case - is the same.

For instance, in the original exercise \(\frac{a^{4}}{a^{6}}\), applying this property gives us \(a^{4-6} = a^{-2}\).

It's a straightforward and incredibly useful property for simplifying algebraic expressions involving exponents, and it's one you'll encounter frequently in algebra.

Properties of Exponents

The properties of exponents are rules that describe how terms with exponents are manipulated. In addition to the quotient of powers property, several other key properties include:

• Product of Powers: When multiplying terms with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).
• Power of a Power: When raising an exponent to another exponent, you multiply the exponents: \( (a^m)^n = a^{m \times n}\).
• Power of a Product: When raising a product to an exponent, you raise each factor to the exponent: \( (ab)^n = a^n b^n\).
• Zero Exponent: Any base (except zero) raised to the zeroth power is one: \(a^0 = 1\).
• Negative Exponents: Indicate the reciprocal of the base raised to the positive exponent: \(a^{-n} = \frac{1}{a^n}\).

Understanding all these properties allows you to manipulate and simplify expressions throughout algebra effectively.

Simplifying Algebraic Expressions

In algebra, simplifying algebraic expressions is the process of making them as basic or as 'clean' as possible. This often means reducing the number of terms, combining like terms, and applying properties of exponents.

To simplify expressions efficiently, follow these steps:

• Combine like terms, which are terms that have the same variable parts.
• Use the distributive property to eliminate parentheses.
• Apply the properties of exponents to simplify powers and roots.

For example, \(a^4 / a^6\) simplifies to \(a^{-2}\) by using the quotient of powers property. You might also encounter more complex expressions where you will need to combine several properties to fully simplify the expression.

Negative Exponents

Negative exponents are not as daunting as they appear. A negative exponent tells us to take the reciprocal of the base and then apply the positive exponent. So, \(a^{-n}\) is the same as \(\frac{1}{a^n}\).

This rule stems from the definition of exponents and particularly from the quotient of powers property. For example, \(a^4 / a^6\) simplifying to \(a^{-2}\) implies that you have \(\frac{1}{a^2}\), since the negative exponent indicates division by the base raised to the positive exponent.

The key takeaway should be that a negative exponent flips the base to its reciprocal. Mastering this concept is crucial for working with exponents in algebra and higher-level mathematics.