Evaluate The Following:$ \begin{array}{l} (-5)^2= \\ (-7)^3= \end{array} $

Evaluating Exponents: A Guide to Understanding the Rules

Exponents are a fundamental concept in mathematics, and understanding how to evaluate them is crucial for solving various mathematical problems. In this article, we will evaluate two expressions involving exponents: $(-5)^2$ and $(-7)^3$. We will delve into the rules governing exponents and provide step-by-step solutions to these expressions.

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication. For example, $a^2$ means $a$ multiplied by itself, i.e., $a \times a$. Similarly, $a^3$ means $a$ multiplied by itself three times, i.e., $a \times a \times a$. Exponents can be positive or negative, and they can also be fractions or decimals.

Evaluating $(-5)^2$

To evaluate $(-5)^2$, we need to follow the rules governing exponents. When an exponent is even, the result is always positive, regardless of the sign of the base. In this case, the exponent is 2, which is even. Therefore, we can ignore the negative sign and simply square the absolute value of the base.

$(-5)^2 = |{-5}|^2 = 5^2 = 25$

So, the value of $(-5)^2$ is 25.

Evaluating $(-7)^3$

To evaluate $(-7)^3$, we need to follow the rules governing exponents. When an exponent is odd, the result is the same as the base raised to that power, regardless of the sign of the base. In this case, the exponent is 3, which is odd. Therefore, we can ignore the negative sign and simply cube the absolute value of the base.

$(-7)^3 = |{-7}|^3 = 7^3 = 343$

So, the value of $(-7)^3$ is 343.

Rules Governing Exponents

There are several rules governing exponents that we need to follow when evaluating expressions involving exponents. These rules are:

• Product Rule: $a^m \times a^n = a^{m+n}$
• Power Rule: $(a^m)^n = a^{mn}$
• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$
• Zero-Exponent Rule: $a^0 = 1$
• Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$

In conclusion, evaluating exponents is a crucial concept in mathematics. By following the rules governing exponents, we can simplify complex expressions and solve various mathematical problems. In this article, we evaluated two expressions involving exponents: $(-5)^2$ and $(-7)^3$. We demonstrated how to apply the rules governing exponents to simplify these expressions and arrive at their respective values.

When evaluating exponents, there are several common mistakes to avoid. These mistakes include:

• Ignoring the sign of the base: When evaluating expressions involving negative bases, it's essential to remember that the sign of the base affects the result.
• Not following the rules governing exponents: Failing to follow the rules governing exponents can lead to incorrect results.
• Not simplifying expressions: Failing to simplify expressions involving exponents can make it difficult to arrive at the correct result.

Exponents have numerous real-world applications in various fields, including:

• Science: Exponents are used to describe the growth or decay of populations, chemical reactions, and physical phenomena.
• Engineering: Exponents are used to describe the behavior of electrical circuits, mechanical systems, and other complex systems.
• Finance: Exponents are used to calculate interest rates, investment returns, and other financial metrics.

In conclusion, evaluating exponents is a fundamental concept in mathematics that has numerous real-world applications. By following the rules governing exponents and avoiding common mistakes, we can simplify complex expressions and solve various mathematical problems. Whether you're a student, a professional, or simply someone interested in mathematics, understanding exponents is essential for success in various fields.