Math Problems: Rational Numbers & Exponents Simplified

Hey guys! Let's dive into some math problems involving rational numbers and exponents. We'll break down each problem step by step so you can ace your math game. Get ready to sharpen those pencils and flex your brain muscles! Let's jump right into it.

13. Multiplying by a Rational Number

Okay, so this problem asks: What rational number should 3' be multiplied by to get a product of 3'? This might seem a bit tricky at first, but let's break it down. The key here is understanding what a "rational number" is and how multiplication works. Essentially, we're looking for a number that, when multiplied by 3, gives us 3.

To solve this, let's use some basic algebra. If we call the rational number we're looking for x, then we can write the problem as an equation:

3 * x = 3

Now, to find x, we need to isolate it. We can do this by dividing both sides of the equation by 3:

(3 * x) / 3 = 3 / 3

This simplifies to:

x = 1

So, the rational number we need to multiply 3 by to get 3 is 1. But hold on a second! Looking at the original problem, it seems like there might be some confusion in the notation. The ' after the 3 might be a typo or meant to represent something else. If it's just a typo, then our answer of 1 is correct. However, if it represents feet (3'), then the problem becomes a bit ambiguous without further context. But assuming it's just the number 3, our answer is definitely 1.

Let's think about why this makes sense. Multiplication by 1 is a fundamental concept in math – any number multiplied by 1 remains the same. This is known as the identity property of multiplication. So, when we multiply 3 by 1, we naturally get 3. This understanding is super important for tackling more complex math problems later on. Think of 1 as the neutral element in multiplication, just like 0 is the neutral element in addition.

Now, let's consider the answer choices provided in the original problem. We had options like 9, 1/9, 3, and 1/3. We can quickly eliminate these by mentally multiplying them by 3. For example, 3 multiplied by 9 gives us 27, which is definitely not 3. Similarly, 3 multiplied by 1/9 gives us 1/3, and 3 multiplied by 3 gives us 9 – neither of which are 3. Only 3 multiplied by 1 gives us 3, confirming our algebraic solution.

In summary, the rational number that 3 should be multiplied by to get 3 is undoubtedly 1. Remember this principle – multiplying a number by 1 doesn't change its value. This is a cornerstone of mathematical operations and will help you solve a variety of problems in the future. Keep this trick in your math toolkit, guys!

14. Evaluating an Expression with Exponents

Alright, let's tackle the next problem: What is the value of $5^{8} + 5^{7} - 5^{6}.3^{3} - 5^{6}.3$? This one looks a bit intimidating with all those exponents, but don't worry, we can break it down using some clever techniques. The key here is to identify common factors and simplify the expression.

First, let’s rewrite the expression to make it a bit clearer:

$5^{8} + 5^{7} - 5^{6}.3^{3} - 5^{6}.3$

Notice that $5^{6}$ appears in two of the terms. This is a huge clue! We can factor out $5^{6}$ from those terms:

$5^{8} + 5^{7} - 5^{6}(3^{3} + 3)$

Now, let's think about the first two terms, $5^{8}$ and $5^{7}$. Can we express them in terms of $5^{6}$ as well? Absolutely! Remember the rules of exponents: $x^{a+b} = x^{a} . x^{b}$. So we can rewrite:

$5^{8} = 5^{6+2} = 5^{6} . 5^{2}$

$5^{7} = 5^{6+1} = 5^{6} . 5^{1}$

Now our expression looks like this:

$5^{6} . 5^{2} + 5^{6} . 5 - 5^{6}(3^{3} + 3)$

See another common factor? That's right, $5^{6}$ is in all the terms now! Let's factor it out completely:

$5^{6}(5^{2} + 5 - (3^{3} + 3))$

This is much simpler! Now we just need to evaluate the expression inside the parentheses. Let's start with the exponents:

$5^{2} = 25$

$3^{3} = 27$

Substitute these values back into the parentheses:

$5^{6}(25 + 5 - (27 + 3))$

Now we perform the additions and subtractions inside the parentheses:

$5^{6}(30 - 30)$

What's 30 - 30? It's 0! So we have:

$5^{6}(0)$

And anything multiplied by 0 is 0. Therefore, the value of the entire expression is 0!

Isn't it cool how we transformed a seemingly complex expression into something so simple? This problem highlights the power of factoring and using exponent rules to simplify expressions. Remember, when you see terms with exponents, look for common factors – it can save you a ton of calculation time. Factoring is your best friend in these kinds of problems, guys! It's like a secret weapon for math ninjas!

15. Simplifying Expressions with Exponents

Let's move on to our final problem: Simplify: $x^{a}.x^{b}.x^{c}.x^{a}.x^{-b}.x^{-c}$ This problem is all about applying the rules of exponents. The key rule we'll use here is the product of powers rule: $x^{m} . x^{n} = x^{m+n}$. This rule states that when you multiply terms with the same base, you add the exponents.

So, let's rewrite our expression, grouping the terms together:

$x^{a}.x^{b}.x^{c}.x^{a}.x^{-b}.x^{-c}$

Now, let's apply the product of powers rule to combine all the x terms. We simply add up all the exponents:

$x^{a + b + c + a + (-b) + (-c)}$

Notice anything interesting in the exponent? We have both positive and negative b and c terms. Let's simplify the exponent by combining like terms:

$x^{a + b + c + a - b - c}$

The b and -b cancel each other out, and the c and -c also cancel out. This leaves us with:

$x^{a + a}$

Which simplifies to:

$x^{2a}$

And that's it! We've successfully simplified the expression to $x^{2a}$. This problem demonstrates how powerful the exponent rules can be. By understanding and applying these rules, you can simplify even the most complicated-looking expressions. The beauty of math, guys, is that seemingly complex problems often have elegant and simple solutions if you know the right tools and techniques.

This particular problem also highlights the importance of paying attention to the signs of the exponents. Negative exponents can sometimes trip people up, but as we saw here, they often lead to cancellations and simplifications. Always double-check your work and make sure you're handling those negatives correctly. It's a small detail that can make a big difference in the final answer. Keep practicing these exponent rules, and you'll become a pro in no time!

So, there you have it! We've tackled three different math problems, each requiring a different set of skills and techniques. Remember, the key to success in math is to break down complex problems into smaller, more manageable steps. Look for patterns, identify common factors, and don't be afraid to use the rules and properties you've learned. And most importantly, practice, practice, practice! The more you work through problems, the more comfortable and confident you'll become. Keep up the great work, and happy math-ing, everyone!