Solving Mathematical Expressions: Simplifying and Explaining Steps

Mathematics often involves breaking complex expressions into simpler components for easier solving. Let's dive into the various expressions and expressions, understand how they work, and provide detailed steps for each calculation.

Adding and Subtracting Integers

Let's start with some basic arithmetic operations involving positive and negative integers.

8 - 5

Imagine you have 8 apples and someone asks you to take away 5 apples. How many apples would you have left? To find this, we simplify the expression:

Starting with 8 apples, take away 5 apples, you are left with 3 apples. Thus, 8 - 5 equals 3.

-1 - 1 - 1 - -1

This is a bit more complex but still manageable. Let's simplify step by step:

-1 - 1 -2 -2 - -1 -1 (Remember, subtracting a negative is the same as adding a positive) -2 - 1 -3 (Incorrect in the given solution, should be -1 1 0) -1 - -1 0 (Same logic as above)

Now, substitute back into the original expression:

-1 - 1 - 1 - -1 -2 - 1 -3

So, -1 - 1 - 1 - -1 equals -3 (Corrected solution).

-7 - 8

To find the result of -7 - 8:

Add the two negative numbers together: -7 - 8 -15. So, the answer is -15.

8 - 5

Again, to solve 8 - 5:

Start with 8 and take away 5, you are left with 3. So, 8 - 5 equals 3.

Working with Exponents

Now, let's tackle an expression involving exponents. We need to simplify the expression:

{7^1/2 8^1/2 ÷ 8^1/2 7^1/2}

First, simplify the numerator and the denominator:

Numerator: 7^1/2 8^1/2 √7 √8 Denominator: 8^1/2 7^1/2 √8 √7

Now, rewrite the expression:

√7 √8 ÷ √8 √7 1 (Since the terms in the numerator and denominator cancel out)

So, the simplified value of the expression is 1.

Multiplication of Negative Numbers

To solve -8 × -9:

The product of two negative numbers is always positive. Thus, -8 × -9 equals 72.

Similarly, -6 × -8 equals 48 since the product of two negative numbers is positive.

Basic Multiplication of Integers

To solve 5 × 10 × 10:

Multiply the three numbers together: 5 × 10 × 10 500. So, the answer is 500.

Pattern Recognition

Understanding patterns in number sequences is crucial in mathematics. Let's analyze the sequence 2, 4, 6, 8, 10.

Pattern Rule of 2, 4, 6, 8, 10

The pattern rule is:

Start with 2. Add 2 to each subsequent term to get the next term in the sequence.

This sequence follows a simple rule: each term is 2 more than the previous term.

Summary

In this article, we have covered a variety of mathematical expressions and their solutions. We have explored operations with integers, simplifying expressions with exponents, and recognizing patterns in number sequences. By breaking down each problem into simpler steps, we can solve complex expressions more easily.