Simplify The Expression: ${ (-10) \cdot (-10)^4 }$

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. When dealing with exponents and multiplication, it's essential to understand the rules and apply them correctly. In this article, we will simplify the expression $(-10) \cdot (-10)^4$ using the properties of exponents and multiplication.

Understanding Exponents

Before we dive into simplifying the expression, let's review the concept of exponents. An exponent is a small number that is written above and to the right of a larger number, indicating how many times the larger number should be multiplied by itself. For example, $a^b$ means $a$ multiplied by itself $b$ times. In the expression $(-10)^4$, the exponent $4$ indicates that $-10$ should be multiplied by itself $4$ times.

Simplifying the Expression

Now that we understand exponents, let's simplify the expression $(-10) \cdot (-10)^4$. To do this, we need to apply the rule of multiplication with exponents. When multiplying two numbers with the same base, we add the exponents. In this case, the base is $-10$, and the exponents are $1$ and $4$. Therefore, we can simplify the expression as follows:

$$(-10) \cdot (-10)^4 = (-10)^{1+4} = (-10)^5$$

Evaluating the Expression

Now that we have simplified the expression to $(-10)^5$, let's evaluate it. To do this, we need to multiply $-10$ by itself $5$ times:

$$(-10)^5 = (-10) \cdot (-10) \cdot (-10) \cdot (-10) \cdot (-10)$$

Multiplying Negative Numbers

When multiplying negative numbers, we need to remember that an even number of negative factors results in a positive product, while an odd number of negative factors results in a negative product. In this case, we have $5$ negative factors, which is an odd number. Therefore, the product will be negative.

Calculating the Product

Now that we know the product will be negative, let's calculate it:

$$(-10) \cdot (-10) \cdot (-10) \cdot (-10) \cdot (-10) = -100000$$

Conclusion

In conclusion, we have simplified the expression $(-10) \cdot (-10)^4$ using the properties of exponents and multiplication. We first simplified the expression to $(-10)^5$ and then evaluated it by multiplying $-10$ by itself $5$ times. The final product is $-100000$.

Frequently Asked Questions

• Q: What is the rule for multiplying numbers with the same base? A: When multiplying two numbers with the same base, we add the exponents.
• Q: What is the rule for multiplying negative numbers? A: An even number of negative factors results in a positive product, while an odd number of negative factors results in a negative product.
• Q: How do we simplify an expression with exponents? A: We apply the rule of multiplication with exponents by adding the exponents.

Final Answer

The final answer is: $\boxed{-100000}$

Introduction

In our previous article, we simplified the expression $(-10) \cdot (-10)^4$ using the properties of exponents and multiplication. In this article, we will answer some frequently asked questions related to simplifying expressions with exponents and multiplication.

Q&A

Q: What is the rule for multiplying numbers with the same base?

A: When multiplying two numbers with the same base, we add the exponents. For example, $a^b \cdot a^c = a^{b+c}$.

Q: What is the rule for multiplying negative numbers?

A: An even number of negative factors results in a positive product, while an odd number of negative factors results in a negative product. For example, $(-a) \cdot (-b) = ab$ and $(-a) \cdot b = -ab$.

Q: How do we simplify an expression with exponents?

A: We apply the rule of multiplication with exponents by adding the exponents. For example, $a^b \cdot a^c = a^{b+c}$.

Q: What is the difference between $(-10)^4$ and $(-10)^{-4}$?

A: $(-10)^4$ means $-10$ multiplied by itself $4$ times, resulting in a positive product. On the other hand, $(-10)^{-4}$ means $-10$ raised to the power of $-4$, which is equivalent to $1/(-10)^4$. This results in a negative product.

Q: Can we simplify an expression with a negative exponent?

A: Yes, we can simplify an expression with a negative exponent by using the rule $a^{-b} = 1/a^b$. For example, $(-10)^{-4} = 1/(-10)^4$.

Q: How do we evaluate an expression with a negative exponent?

A: To evaluate an expression with a negative exponent, we need to follow the order of operations (PEMDAS). We first simplify the expression inside the parentheses and then apply the rule for negative exponents.

Q: Can we simplify an expression with a zero exponent?

A: Yes, we can simplify an expression with a zero exponent by using the rule $a^0 = 1$. For example, $(-10)^0 = 1$.

Q: How do we evaluate an expression with a zero exponent?

A: To evaluate an expression with a zero exponent, we simply apply the rule $a^0 = 1$. For example, $(-10)^0 = 1$.

Conclusion

In conclusion, we have answered some frequently asked questions related to simplifying expressions with exponents and multiplication. We have covered topics such as multiplying numbers with the same base, multiplying negative numbers, simplifying expressions with exponents, and evaluating expressions with negative and zero exponents.

Final Answer

The final answer is: $\boxed{-100000}$