algebra concepts chapter 7 introduction to exponents mastery check
1. 7.1 evaluate $-6^1$
2. 7.2 simplify $x^8 \cdot x$
3. 7.2 simplify $t^3 u^8 \cdot t^3$
4. 7.3 simplify $2x \cdot 6x^3$
5. 7.3 simplify $5b^4 c^8 \cdot b c^3$
6. 7.4 simplify $3n(7n + 6)$

Question

Accepted Answer

### Problem 1: Evaluate $-6^1$
# Explanation:
## Step1: Recall exponent rule for $a^1$
Any number to the power of 1 is itself, so $6^1 = 6$. The negative sign is outside the exponentiation, so we apply it to the result of $6^1$.
## Step2: Calculate the value
$-6^1=- (6^1)= - 6$
# Answer: $-6$

### Problem 2: Simplify $x^8\cdot x$
# Explanation:
## Step1: Recall the product of powers rule
The rule is $a^m\cdot a^n=a^{m + n}$, where $a$ is a non - zero real number and $m,n$ are integers. Here, $a = x$, $m = 8$ and $n=1$ (since $x=x^1$).
## Step2: Apply the rule
$x^8\cdot x=x^{8 + 1}=x^9$
# Answer: $x^9$

### Problem 3: Simplify $t^3u^8\cdot t^3$
# Explanation:
## Step1: Recall the product of powers rule for like bases
For the base $t$, we use the rule $a^m\cdot a^n=a^{m + n}$. The base $u$ has an exponent of 8 and since there is no other $u$ term, its exponent remains 8.
## Step2: Apply the rule to the $t$ terms
For the $t$ terms, $m = 3$ and $n = 3$. So $t^3\cdot t^3=t^{3+3}=t^6$. The $u^8$ term stays as it is. So the simplified form is $t^6u^8$
# Answer: $t^6u^8$

### Problem 4: Simplify $2x\cdot6x^3$
# Explanation:
## Step1: Multiply the coefficients and use the product of powers rule for $x$ terms
First, multiply the coefficients: $2\times6 = 12$. Then, for the $x$ terms, use the rule $a^m\cdot a^n=a^{m + n}$ with $a=x$, $m = 1$ and $n = 3$.
## Step2: Combine the results
$x^1\cdot x^3=x^{1 + 3}=x^4$. Then multiply by the coefficient 12. So $2x\cdot6x^3=12x^4$
# Answer: $12x^4$

### Problem 5: Simplify $5b^4c^8\cdot bc^3$
# Explanation:
## Step1: Multiply the coefficients and apply product of powers rule to $b$ and $c$ terms
- Coefficient: $5\times1 = 5$
- For $b$ terms: $b^4\cdot b=b^{4 + 1}=b^5$ (using $a^m\cdot a^n=a^{m + n}$ with $a = b$, $m = 4$, $n = 1$)
- For $c$ terms: $c^8\cdot c^3=c^{8+3}=c^{11}$ (using $a^m\cdot a^n=a^{m + n}$ with $a = c$, $m = 8$, $n = 3$)
## Step2: Combine the results
Multiply the coefficient, $b$ term, and $c$ term together. So $5b^4c^8\cdot bc^3 = 5b^5c^{11}$
# Answer: $5b^5c^{11}$

### Problem 6: Simplify $3n(7n + 6)$
# Explanation:
## Step1: Use the distributive property $a(b + c)=ab+ac$
Here, $a = 3n$, $b = 7n$ and $c = 6$.
## Step2: Apply the distributive property
- Multiply $3n$ and $7n$: $3n\times7n=21n^2$ (using $a^m\cdot a^n=a^{m + n}$ with $a = n$, $m = 1$, $n = 1$ and multiplying the coefficients $3\times7 = 21$)
- Multiply $3n$ and $6$: $3n\times6 = 18n$
## Step3: Combine the terms
So $3n(7n + 6)=21n^2+18n$
# Answer: $21n^2 + 18n$

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QUESTION IMAGE

Question

Explanation:

Problem 1: Evaluate \(-6^1\)

Step1: Recall exponent rule for \(a^1\)

Step2: Calculate the value

Step1: Recall the product of powers rule

Step2: Apply the rule

Step1: Recall the product of powers rule for like bases

Step2: Apply the rule to the \(t\) terms

Answer:

Problem 2: Simplify \(x^8\cdot x\)