Rules of Roots Combined Practice Problems & Exercises

We use the zero exponent rule.

$X^0=1$We obtain

$112^0=1$Therefore, the correct answer is option C.

In order to solve this problem, we'll follow these steps:

• Step 1: Identify the base and exponents

• Step 2: Use the formula for multiplying powers with the same base

• Step 3: Simplify the expression by applying the relevant exponent rule

Now, let's work through each step:

Step 1: The given expression is $(3^4) \times (3^2)$. Here, the base is 3, and the exponents are 4 and 2.

Step 2: Apply the exponent rule, which states that when multiplying powers with the same base, we add the exponents:
$a^m \times a^n = a^{m+n}$

Step 3: Using the rule identified in Step 2, we add the exponents 4 and 2:
$3^4 \times 3^2 = 3^{4+2} = 3^6$

Therefore, the simplified form of the expression is $3^6$.

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 $4^5 \times 3^7$.

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

• Step 1: Identify and group the terms with the same base.

• Step 2: Apply the laws of exponents to simplify by adding the exponents of each base.

• Step 3: Write the simplified form.

Let's work through each step:

Step 1: We are given that $4^7 \times 5^3 \times 4^2 \times 5^4$.

Step 2: First, group the terms with the same base:

$4^7 \times 4^2$ and $5^3 \times 5^4$.

Step 3: Use the law of exponents, which states $a^m \times a^n = a^{m+n}$.

For the base 4: $4^7 \times 4^2 = 4^{7+2} = 4^9$.

For the base 5: $5^3 \times 5^4 = 5^{3+4} = 5^7$.

Therefore, the simplified form of the expression is $4^9 \times 5^7$.