Problem 15 - Elementary Algebra

Question

Find each quotient $$ \frac{a^{7}}{a} $$

Step-by-Step Solution

Answer

Answer: The quotient of the expression $$\frac{a^{7}}{a}$$ is $$a^6$$.

Step 1: Identify the Exponents

In the given expression, the base is 'a' and the exponents are 7 and 1 respectively: $$a^7$$ and $$a^1$$.

Step 2: Apply Quotient Rule for Exponents

According to the quotient rule, subtract the exponent in the denominator from the exponent in the numerator with the same base: $$a^m \div a^n = a^{m-n}$$. In this case, $$a^7 \div a^1 = a^{7-1} $$.

Step 3: Simplify the Expression

Subtract the exponents as shown in the previous step to arrive at the simplified expression: $$a^{7-1} = a^6$$. The quotient of the given expression $$\frac{a^{7}}{a}$$ is $$a^6$$.

Key Concepts

Understanding Exponents

Exponents provide a way to express repeated multiplication of a number by itself. In the expression \(a^7\), "7" is the exponent, and it tells us how many times "a" is multiplied by itself. This is read as "a raised to the power of 7." Similarly, in \(a^1\), the exponent is 1, which means "a" is used just once, essentially, it remains "a."

• The base "a": The number being multiplied.
• Exponent 7: The number of times "a" appears in the multiplication.
• Exponent 1: Indicates the number "a" exists singularly.

By understanding these basic components, you'll see how exponents simplify longer expressions and represent large values succinctly.

The Art of Simplification

Simplification is the process of making an expression easier to work with while maintaining its original value. When it comes to expressions with exponents, simplification often involves applying certain rules like the quotient rule.

In our given expression \(\frac{a^{7}}{a}\), simplification involves reducing the complexity by handling the exponents. When you apply the quotient rule, you focus on the exponents of the same base to perform subtraction. Here it means \(a^{7-1}\). It is crucial because it turns a potentially confusing fraction into a straightforward expression, \(a^6\), which is much easier to understand and use. Remember, simpler forms are not just neat; they are essential for further operations.

Algebraic Expressions and Their Simplifications

Algebraic expressions can have variables, constants, and exponents. In any simplified algebraic expression like the one found here, understanding the role of each part is key.

Consider the original expression \(\frac{a^{7}}{a}\). It is an algebraic expression composed of a single variable "a" raised to different powers. To simplify such expressions, you use algebraic rules and properties, including the exponent rules.

• Quotient Rule: For same base division, subtract exponents: \(a^m \div a^n = a^{m-n}\).
• Result: Fewer terms and easier calculations, like going from \(a^{7}\) divided by \(a\) to \(a^6\).

These simplifications are crucial for solving larger algebraic expressions efficiently and accurately, ensuring you keep expressions manageable.