Problem 2 - Elementary Algebra

Question

Simplify each expression using the power rule for powers. $$ \left(y^{7}\right)^{7} $$

Step-by-Step Solution

Answer

Answer: \(y^{49}\)

Step 1: Identify the base and exponents

In this expression, the base is y, and it has an exponent of 7. The entire expression is then raised to the power of 7.

Step 2: Apply the power rule for powers

According to the power rule for powers, when a power is raised to another power, we multiply the exponents together. To simplify the given expression, multiply the two exponents: $$ (y^{7})^{7} = y^{(7 \cdot 7)} $$

Step 3: Calculate the new exponent

Multiply the two exponents 7 and 7 together to get the new exponent for y: $$ 7 \cdot 7 = 49 $$

Step 4: Write the simplified expression

Now that we have calculated the new exponent, write the simplified expression with the new exponent: $$ y^{(7\cdot7)}=y^{49} $$ So, \(\left(y^{7}\right)^{7}\) simplifies to \(y^{49}\).

Key Concepts

Understanding Exponents

Exponents are a way of expressing repeated multiplication of the same number or variable. In an expression like \(a^n\), \(a\) is the base and \(n\) is the exponent. The exponent tells you how many times to multiply the base by itself. For example, if you have \(2^3\), it means \(2 \times 2 \times 2\), which equals 8.
When dealing with exponents, it's essential to understand the power rule for powers, which applies when you have a power raised to another power. In our exercise, \((y^7)^7\), the base is \(y\) with an initial exponent of 7, which is then raised to another power of 7. This is where the power rule comes in handy.

Simplifying Expressions with the Power Rule

Simplifying mathematical expressions means expressing them in a simpler or more concise form without changing their value. When simplifying expressions with exponents, one of the most powerful tools is the power rule for powers.
The power rule states that when you have an exponentiation of an exponent, or a power raised to a power, you multiply the exponents. For example, in the problem \((y^7)^7\), you multiply the exponents 7 and 7 together. This gives us a new expression where the base remains the same, but the exponent becomes the product of the two exponents: \(y^{49}\).
This rule simplifies the process and makes expressions easier to work with because it reduces the number of operations needed in evaluations or further algebraic manipulations.

Algebra and Exponent Rules

Algebra is the branch of mathematics that uses symbols (like \(x\) or \(y\)) to represent numbers in equations and expressions. It provides the rules and techniques needed to solve these equations and rearrange expressions into simpler forms.
Exponents play a crucial role in algebra because they allow for concise representation of multiplication, especially when the multiplication is repeated several times. Knowing how to manipulate expressions with exponents is crucial for solving algebraic problems efficiently.

• The basic exponent rules - including the product, quotient, and power rules - are foundational tools in algebra.
• The product rule states that when multiplying two terms with the same base, you add the exponents: \(a^m \times a^n = a^{m+n} \).
• The quotient rule tells us to subtract the exponents of like bases when dividing: \(\frac{a^m}{a^n} = a^{m-n}\).
• The power rule, as we used in our exercise, simplifies expressions where a power is raised to another power by multiplying the exponents.

Mastering these rules will not only help in simplifying complex expressions but also in solving more advanced algebraic equations with ease.