Break down the numbers into their prime factors

Oct 13, 2025 by leolavanderia.com.br 121 views

Introduction

Simplifying complex expressions is an essential skill in mathematics, particularly in algebra and calculus. It involves reducing complex expressions to their simplest form, making it easier to solve equations and inequalities. In this article, we will focus on simplifying four complex expressions without using a calculator.

Simplifying Expression a)

Expression a)

$\frac{20^3 \times 25^2 \times 4}{10^7}$

Simplifying Expression a)

To simplify this expression, we need to apply the rules of exponents. We can start by breaking down the numbers into their prime factors.

$20 = 2^2 \times 5$ $25 = 5^2$ $10 = 2 \times 5$

Now, we can rewrite the expression using these prime factors.

$\frac{(2^2 \times 5)^3 \times (5^2)^2 \times 2^2}{(2 \times 5)^7}$

Next, we can apply the power of a product rule, which states that $(ab)^n = a^n \times b^n$ . We can also apply the power of a power rule, which states that $(a^m)^n = a^{m \times n}$ .

$\frac{2^6 \times 5^6 \times 5^4 \times 2^2}{2^7 \times 5^7}$

Now, we can simplify the expression by combining like terms.

$\frac{2^8 \times 5^{10}}{2^7 \times 5^7}$

Finally, we can apply the quotient of powers rule, which states that $\frac{a^m}{a^n} = a^{m-n}$ . We can also apply the product of powers rule, which states that $a^m \times a^n = a^{m+n}$ .

$\frac{2^1 \times 5^3}{1}$

$\boxed{2 \times 125}$

Answer

The simplified expression is $2 \times 125 = 250$ .

Simplifying Expression b)

Expression b)

$\frac{10^3 \cdot 15^4 \cdot 6^2}{20^4 \cdot 12 \cdot 45^2}$

Simplifying Expression b)

To simplify this expression, we need to apply the rules of exponents. We can start by breaking down the numbers into their prime factors.

$10 = 2 \times 5$ $15 = 3 \times 5$ $6 = 2 \times 3$ $20 = 2^2 \times 5$ $12 = 2^2 \times 3$ $45 = 3^2 \times 5$

Now, we can rewrite the expression using these prime factors.

$\frac{(2 \times 5)^3 \cdot (3 \times 5)^4 \cdot (2 \times 3)^2}{(2^2 \times 5)^4 \cdot (2^2 \times 3) \cdot (3^2 \times 5)^2}$

Next, we can apply the power of a product rule, which states that $(ab)^n = a^n \times b^n$ . We can also apply the power of a power rule, which states that $(a^m)^n = a^{m \times n}$ .

$\frac{2^3 \times 5^3 \cdot 3^4 \times 5^4 \cdot 2^2 \times 3^2}{2^8 \times 5^4 \cdot 2^2 \times 3 \cdot 3^4 \times 5^4}$

Now, we can simplify the expression by combining like terms.

$\frac{2^5 \times 3^6 \times 5^7}{2^10 \times 3^5 \times 5^8}$

Finally, we can apply the quotient of powers rule, which states that $\frac{a^m}{a^n} = a^{m-n}$ . We can also apply the product of powers rule, which states that $a^m \times a^n = a^{m+n}$ .

$\frac{2^{-5} \times 3 \times 5^{-1}}{1}$

$\boxed{\frac{3}{2^5 \times 5}}$

Answer

The simplified expression is $\frac{3}{2^5 \times 5} = \frac{3}{1000}$ .

Simplifying Expression c)

Expression c)

$\frac{2 \times 2^{2a} \times 16^2}{4^{a+5}}$

Simplifying Expression c)

To simplify this expression, we need to apply the rules of exponents. We can start by breaking down the numbers into their prime factors.

$2 = 2$ $16 = 2^4$ $4 = 2^2$

Now, we can rewrite the expression using these prime factors.

$\frac{2 \times 2^{2a} \times (2^4)^2}{(2^2)^{a+5}}$

Next, we can apply the power of a product rule, which states that $(ab)^n = a^n \times b^n$ . We can also apply the power of a power rule, which states that $(a^m)^n = a^{m \times n}$ .

$\frac{2 \times 2^{2a} \times 2^8}{2^{2a+10}}$

Now, we can simplify the expression by combining like terms.

$\frac{2^{2a+10}}{2^{2a+10}}$

Finally, we can apply the quotient of powers rule, which states that $\frac{a^m}{a^n} = a^{m-n}$ . We can also apply the product of powers rule, which states that $a^m \times a^n = a^{m+n}$ .

$\frac{1}{1}$

$\boxed{1}$

Answer

The simplified expression is $1$ .

Simplifying Expression d)

Expression d)

$\frac{3^2 \cdot 4^3 \cdot 9^4}{3^5 \cdot 8 \cdot 27^2}$

Simplifying Expression d)

To simplify this expression, we need to apply the rules of exponents. We can start by breaking down the numbers into their prime factors.

$3 = 3$ $4 = 2^2$ $9 = 3^2$ $8 = 2^3$ $27 = 3^3$

Now, we can rewrite the expression using these prime factors.

$\frac{(3^2)^2 \cdot (2^2)^3 \cdot (3^2)^4}{3^5 \cdot 2^3 \cdot (3^3)^2}$

Next, we can apply the power of a product rule, which states that $(ab)^n = a^n \times b^n$ . We can also apply the power of a power rule, which states that $(a^m)^n = a^{m \times n}$ .

$\frac{3^4 \cdot 2^6 \cdot 3^8}{3^5 \cdot 2^3 \cdot 3^6}$

Now, we can simplify the expression by combining like terms.

$\frac{3^{-1} \cdot 2^3}{1}$

$\boxed{\frac{8}{3}}$

Answer

The simplified expression is $\frac{8}{3}$ .

Conclusion

Introduction

Simplifying complex expressions is an essential skill in mathematics, particularly in algebra and calculus. In our previous article, we simplified four complex expressions without using a calculator. In this article, we will answer some frequently asked questions about simplifying complex expressions.

Q: What are the rules of exponents?

A: The rules of exponents are a set of rules that govern the behavior of exponents in mathematical expressions. The main rules of exponents are:

Power of a product rule: $(ab)^n = a^n \times b^n$
Power of a power rule: $(a^m)^n = a^{m \times n}$
Quotient of powers rule: $\frac{a^m}{a^n} = a^{m-n}$
Product of powers rule: $a^m \times a^n = a^{m+n}$

Q: How do I simplify a complex expression?

A: To simplify a complex expression, you need to follow these steps:

Break down the numbers into their prime factors: Break down the numbers in the expression into their prime factors.
Apply the power of a product rule: Apply the power of a product rule to simplify the expression.
Apply the power of a power rule: Apply the power of a power rule to simplify the expression.
Apply the quotient of powers rule: Apply the quotient of powers rule to simplify the expression.
Apply the product of powers rule: Apply the product of powers rule to simplify the expression.
Combine like terms: Combine like terms to simplify the expression.

Q: What is the difference between a prime factor and a composite number?

A: A prime factor is a prime number that divides a composite number. A composite number is a number that can be divided by a prime number. For example, the prime factors of 12 are 2 and 3, and the composite number 12 can be divided by 2 and 3.

Q: How do I simplify an expression with negative exponents?

A: To simplify an expression with negative exponents, you need to apply the quotient of powers rule. For example, if you have the expression $\frac{a^m}{a^n}$ , you can simplify it by applying the quotient of powers rule: $\frac{a^m}{a^n} = a^{m-n}$ .

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

A: Yes, you can simplify an expression with a zero exponent. If you have an expression with a zero exponent, such as $a^0$ , you can simplify it by applying the rule that $a^0 = 1$ .

Q: How do I simplify an expression with a fractional exponent?

A: To simplify an expression with a fractional exponent, you need to apply the power of a power rule. For example, if you have the expression $(a^m)^n$ , you can simplify it by applying the power of a power rule: $(a^m)^n = a^{m \times n}$ .

Conclusion

Simplifying complex expressions is an essential skill in mathematics, particularly in algebra and calculus. By applying the rules of exponents, we can simplify complex expressions and make them easier to solve. In this article, we have answered some frequently asked questions about simplifying complex expressions. We hope that this article has been helpful in understanding the rules of exponents and how to simplify complex expressions.