Rules of Roots Combined - Examples, Exercises and Solutions

Let's begin by calculating the numerical value of each of the roots in the given options:

$\sqrt{25}=5\\ \sqrt{16}=4\\ \sqrt{9}=3\\$We can determine that:

5>4>3>1 Therefore, the correct answer is option A

To solve this problem, we'll follow these steps:

• Step 1: Identify the provided expression: $(9^2)^4$.

• Step 2: Apply the power of a power rule for exponents.

• Step 3: Simplify by multiplying the exponents.

Now, let's work through each step:

Step 1: We have the expression $(9^2)^4$.

Step 2: Using the power of a power rule ($(a^m)^n = a^{m \cdot n}$), apply it to the expression:

$(9^2)^4 = 9^{2 \times 4}$

Step 3: Simplify by calculating the product of the exponents:

$2 \times 4 = 8$

Therefore, $(9^2)^4 = 9^8$.

The correct expression corresponding to the given problem is $\boxed{9^8}$.

To solve this problem, we'll follow these steps:

• Identify the given information and relevant exponent rules.

• Apply the quotient property of exponents.

• Simplify the expression.

Now, let's work through each step:
Step 1: The problem gives us the expression $\frac{6^7}{6^4}$. The base is 6, and the exponents are 7 and 4, respectively.
Step 2: According to the rule of exponents, when dividing powers with the same base, we subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$ In this case, $a = 6$, $m = 7$, and $n = 4$.
Step 3: Applying this rule gives us: $\frac{6^7}{6^4} = 6^{7 - 4} = 6^3$

Therefore, the solution to the problem is $6^3$.

To solve this problem, we need to simplify the expression $\frac{b^5}{b^2}$ using the rules of exponents.

• Step 1: Identify the rule to apply: For any positive integer exponents $m$ and $n$, the rule $\frac{a^m}{a^n} = a^{m-n}$ applies when dividing terms with the same base. In this expression, our base is $b$.

• Step 2: Apply the rule: Substitute the given exponents into the formula: $\frac{b^5}{b^2} = b^{5-2}$

• Step 3: Perform the subtraction: Calculate the exponent $5 - 2$: $b^{5-2} = b^3$

Therefore, the solution to the expression $\frac{b^5}{b^2}$ is $b^3$.

To solve the given expression $\frac{x^6}{x^4}$, we will follow these steps:

• Step 1: Apply the quotient rule for exponents
• Step 2: Simplify the expression
• Step 3: Verify by comparing with the answer choices

Now, let's work through each step:

Step 1: Apply the quotient rule for exponents. This rule states that $\frac{a^m}{a^n} = a^{m-n}$ when dividing powers with the same base.

Step 2: We have $\frac{x^6}{x^4}$. According to the rule:

$\frac{x^6}{x^4} = x^{6-4} = x^2$

Step 3: Verify by comparing with the answer choices:

• Choice 1: $x^{-2}$ – Incorrect as it implies the exponents were added incorrectly.
• Choice 2: $x^2$ – This matches our result.
• Choice 3: $x^{10}$ – Incorrect as it implies the exponents were added instead of subtracted.
• Choice 4: $x^{\frac{2}{3}}$ – Incorrect as it does not match the calculation based on integer exponents.

Therefore, the correct choice is $x^2$, which is Choice 2.

To solve the expression $\frac{y^9}{y^3}$, we will apply the rules of exponents, specifically the power of division rule, which states that when you divide like bases, you subtract the exponents.

Here are the steps to arrive at the solution:

• Step 1: Identify and write down the expression: $\frac{y^9}{y^3}$.

• Step 2: Apply the division rule of exponents, which is $\frac{a^m}{a^n} = a^{m-n}$, for any non-zero base $a$.

• Step 3: Using the division rule, subtract the exponent in the denominator from the exponent in the numerator:$y^{9-3}$

• Step 4: Calculate the exponent: $9 - 3 = 6$

• Step 5: Write down the simplified expression:$y^6$

Therefore, the expression $\frac{y^9}{y^3}$ simplifies to $y^6$.

To simplify the expression $(x^3)^4$, we'll follow these steps:

• Step 1: Identify the expression: $(x^3)^4$.
• Step 2: Apply the formula for a power raised to another power.
• Step 3: Calculate the product of the exponents.

Now, let's work through each step:

Step 1: We have the expression $(x^3)^4$, which involves a power raised to another power.

Step 2: We apply the exponent rule $(a^m)^n = a^{m \cdot n}$ here with $a = x$, $m = 3$, and $n = 4$.

Step 3: Multiply the exponents: $3 \times 4 = 12$. This gives us a new exponent for the base $x$.

Therefore, $(x^3)^4 = x^{12}$.

Consequently, the correct answer choice is: $x^{12}$ from the options provided. The other options $x^6$, $x^1$, and $x^7$ do not reflect the correct application of the exponent multiplication rule.

To reduce the expression $a^2 \times a^5 \times a^3$, we will apply the product of powers property of exponents. This property states that when multiplying expressions with the same base, we add their exponents.

• Step 1: Identify the exponents.
  The expression involves the same base $a$ with exponents: 2, 5, and 3.
• Step 2: Add the exponents.
  According to the product of powers property, $a^2 \times a^5 \times a^3 = a^{2+5+3}$.
• Step 3: Simplify the expression.
  Calculate the sum of the exponents: $2 + 5 + 3 = 10$. Therefore, the expression simplifies to $a^{10}$.

Ultimately, the solution to the problem is $a^{10}$. Among the provided choices, is correct: $a^{10}$. The other options $a^5$, $a^8$, and $a^4$ do not correctly reflect the sum of the exponents as calculated.

To solve this problem, we'll employ the power of a power rule in exponents, which states that $(a^m)^n = a^{m \times n}$.

Let's apply this rule to each part of the expression:

• Step 1: Simplify $(3^2)^4$
  According to the power of a power rule, this becomes $3^{2 \times 4} = 3^8$.

• Step 2: Simplify $(5^3)^5$
  Similarly, apply the rule here to get $5^{3 \times 5} = 5^{15}$.

After simplifying both parts, we multiply the results:

$3^8 \times 5^{15}$

Thus, the reduced expression is $\boxed{3^8 \times 5^{15}}$.

To simplify the given expression $4^2 \times 3^5 \times 4^3 \times 3^2$, we will follow these steps:

• Step 1: Identify and group similar bases.

• Step 2: Apply the rule for multiplying like bases.

• Step 3: Simplify the expression.

Now, let's go through each step thoroughly:

Step 1: Identify and group similar bases:
We see two distinct bases here: 4 and 3.

Step 2: Apply the rule for multiplying like bases:
For base 4: Combine $4^2$ and $4^3$, using the rule $a^m \times a^n = a^{m+n}$.

Add the exponents for base 4: $2 + 3 = 5$, thus, $4^2 \times 4^3 = 4^5$.

For base 3: Combine $3^5$ and $3^2$, still using the same exponent rule.

Add the exponents for base 3: $5 + 2 = 7$, resulting in $3^5 \times 3^2 = 3^7$.

Step 3: Simplify the expression:
The simplified expression is $4^5 \times 3^7$.

Therefore, the final simplified expression is $4^5 \times 3^7$.

Let's simplify the expression $5^3 \times 2^4 \times 5^2 \times 2^3$ using the rules for exponents. We'll apply the product of powers rule, which states that when multiplying like bases, you can add the exponents.

• Step 1: Focus on terms with the same base.
  Combine $5^3$ and $5^2$. Since both terms have the base $5$, we apply the rule $a^m \times a^n = a^{m+n}$: $5^3 \times 5^2 = 5^{3+2} = 5^5$

• Step 2: Combine $2^4$ and $2^3$. Similarly, for the base $2$: $2^4 \times 2^3 = 2^{4+3} = 2^7$

After simplification, the expression becomes:
$5^5 \times 2^7$

To simplify the equation $6^4 \times 2^3 \times 6^2 \times 2^5$, we will make use of the rules of exponents, specifically the product of powers rule, which states that when multiplying two powers that have the same base, you can add their exponents.

Step 1: Identify and group the terms with the same base.
In the expression $6^4 \times 2^3 \times 6^2 \times 2^5$, group the powers of 6 together and the powers of 2 together:

• Powers of 6: $6^4 \times 6^2$

• Powers of 2: $2^3 \times 2^5$

Step 2: Apply the product of powers rule.
According to the product of powers rule, for any real number $a$, and integers $m$ and $n$, the expression $a^m \times a^n = a^{m+n}$.

Apply this rule to the powers of 6:
$6^4 \times 6^2 = 6^{4+2} = 6^6$.

Apply this rule to the powers of 2:
$2^3 \times 2^5 = 2^{3+5} = 2^8$.

Step 3: Write down the final expression.
Combining our results gives the simplified expression: $6^6 \times 2^8$.

Therefore, the solution to the problem is $6^6 \times 2^8$.

To solve this problem, we'll use the product of powers property which states $a^m \times a^n = a^{m+n}$.

• Step 1: Simplify the expression by grouping the like bases. The original expression is $7^3 \times 5^2 \times 7^4 \times 5^3$.

• Step 2: Combine the exponents for each base. For base 7: $7^3 \times 7^4 = 7^{3+4} = 7^7$. For base 5: $5^2 \times 5^3 = 5^{2+3} = 5^5$.

• Step 3: Write the simplified expression. After combining the exponents, the expression becomes $7^7 \times 5^5$.

Thus, the solution to the problem is $7^7 \times 5^5$.

To solve this problem, we'll apply the laws of exponents to simplify the expression $7^5 \times 2^3 \times 7^2 \times 2^4$.

Let's follow these steps:

• Step 1: Identify like bases.
  We have two like bases in the expression: 7 and 2.

• Step 2: Apply the product of powers rule for each base separately.
  For the base 7: $7^5 \times 7^2 = 7^{5+2} = 7^7$.
  For the base 2: $2^3 \times 2^4 = 2^{3+4} = 2^7$.

• Step 3: Combine the results.
  The expression simplifies to $7^7 \times 2^7$.

The simplified form of the original expression is therefore $7^7 \times 2^7$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given information

• Step 2: Apply the appropriate exponent rule

• Step 3: Perform the calculation

Now, let's work through each step:
Step 1: The problem gives us the expression $1^3$. This means we have a base of 1 and an exponent of 3.
Step 2: We'll use the exponentiation rule, which states that $a^n = a \times a \times \ldots \times a$ (n times).
Step 3: Since our base is 1, raising 1 to any power will still result in 1. Therefore, we can express this as $1 \times 1 \times 1 = 1$.

Therefore, the solution to $1^3$ is $1$.