Problem 34 - Elementary Algebra

Question

Simplify. $$ (-13) 2-(-23) 3 $$

Step-by-Step Solution

Answer

The expression simplifies to 43.

Step 1: Distribute the negative sign

To simplify the expression, start by distributing the negative signs in the expression. Re-write it as follows: \(-13 \times 2 - (-23 \times 3)\) This will separate the operations into two distinct multiplications.

Step 2: Solve the multiplication

Now, perform the multiplication for each part separately.- Calculate: \(-13 \times 2 = -26\).- Calculate: \(-23 \times 3 = -69\).

Step 3: Simplify the expression

Substitute the products from Step 2 back into the expression:\(-26 - (-69)\).When you subtract a negative, it is the same as adding the positive number, so we rewrite it as \(-26 + 69\).

Step 4: Perform the addition

Add the two numbers obtained:\(-26 + 69 = 43\).

Key Concepts

Distributing Negative Signs

When simplifying expressions in algebra, distributing negative signs is a crucial step. In the given problem, we see the expression

• $$(-13) 2 - (-23) 3$$

The negative sign before the term

• $$-23$$

indicates that we need to "distribute" this negative sign across the term. Distributing a negative sign means flipping the sign of each term within the parentheses it affects. Thus, in the case of subtraction:

• Subtracting $$-23$$ becomes adding $$23$$ because two negatives make a positive.

By distributing the negative signs properly, you prevent errors in sign changes later in the calculation.

Multiplication of Integers

Multiplying integers can involve positive or negative numbers. The rule for multiplication of integers states:

• When multiplying a positive integer by a negative integer, the result is negative.
• When both integers are negative, their product is positive.

In the exercise, we handle two multiplications separately:

• Calculate $$-13 imes 2$$: This results in $$-26$$ because a negative times a positive gives a negative.
• Calculate $$-23 imes 3$$: This results in $$-69$$ following the same rule.

These results will be used to further simplify the original expression. Be attentive to the signs while multiplying, as errors with signs can lead to incorrect answers.

Subtracting Negative Numbers

In mathematics, subtracting a negative number is an interesting operation and often a point of confusion. When you subtract a negative number, you actually end up adding its positive equivalent. For example, the expression:

• $$-26 - (-69)$$

can be tricky at first glance. Here’s the rule:

• Subtracting $$-69$$ is the same as adding $$69$$.
• This is because subtracting a negative results in moving to the right on the number line – or increasing in value.

Therefore,

• $$-26 - (-69)$$

transforms into:

• $$-26 + 69$$.

Understanding this can make operations with integers much smoother.

Addition of Integers

The addition of integers follows straightforward rules, but can sometimes get tricky when dealing with negatives. Let’s break down the problem at hand, given the conversion from the previous section:

• Adding $$-26$$ and $$69$$ involves looking at the signs:

• If both integers were positive, simply add their values.
• If one integer is negative and the other is positive (like here), subtract the smaller absolute value from the larger one and use the sign of the integer with the greater absolute value.

So, for

• $$-26 + 69$$:
• Subtract $$26$$ from $$69$$ to get $$43$$.
• Since $$69$$ has the greater absolute value, the result is positive: $$43$$.

Knowing these rules will help make integer arithmetic more intuitive and prevent mistakes.