Chapter 3.2, Problem 20E

To simplify: ${\left(3a^5\right)}^3$

   $27a^{15}$

Given information: ${\left(3a^5\right)}^3$.

Calculation:

We have ${\left(3a^5\right)}^3$

Take each factor to the third power.

   $3^3{\left(a^5\right)}^3$

Use the Power Property.

   $27a^{15}$

Thus, ${\left(3a^5\right)}^3=27a^{15}$

To find:greatest common factor from $6p^2+6p$

The GCF of $6p^2+6p$is6p.

Given information: $6p^2+6p$.

Calculation:

We have $6p^2+6p$.

Find the GCF of all the terms of the polynomial. Rewrite each term as a product using the GCF. Use the Distributive Property ‘in reverse’ to factor the expression.

   $\begin{array}{l}\underset{\_}{\begin{array}{c}6p^2=2\cdot 3\cdot p\cdot p\\ 6p=2\cdot 3\cdot p\end{array}}\\ \text{GCF}=2\cdot 3\cdot p\end{array}$

   $6p^2+6p=6p\left(p+1\right)$

Thus, the GCF of $6p^2+6p$is6p.

To simplify:the given expression $\sqrt{196}$

$\sqrt{196}$

$=$

$\boxed{14}$

Given information: $\sqrt{196}$

Formula Used:

Exponent rule

$\sqrt{a^m}=a^{\frac{m}{2}}$

Calculation:

Steps

Description

$\sqrt{196}=\sqrt{{14}^2}={14}^{2/2}={14}^1=14$ Rewrite the 196 as product of prime factors and then apply the exponent rule

Conclusion:

After simplifying the given expression $\sqrt{196}$using the exponent rule, we get: $\boxed{14}$

To determine:The correct technician among the two i.e. A and B who are discussing on the correct tire of a vehicle.

The correct answer is option (a) i.e. Technician A.

Explanation:

The statement given by technician A, is correct, as the replacement of a tire with downgraded temperature rating is not recommended which means replacing tire of rating B temperature resistance tire with A graded one will improve tire life and quality of ride. Hence, technician A is correct.

The statement given by technician B, is not correct, as the traction rating indicates the ability of the tire to stop on surface and not traction of tire on snow. Hence, technician B is not correct.

Conclusion:

Thus, the statement given by technician A is correct.

7. Express these system specifications using the propositionsp:“The message is scanned for viruses” andq:“The message was sent from an unknown system” together with logical connectives (including negations).

a) "The message is scanned for viruses whenever the message was sent from an unknown system."

b) "The message was sent from an unknown system but it was not scanned for viruses."

c) "It is necessary to scan the message for viruses whenever it was sent from an unknown system."

d) "When a message is not sent from an unknown system it is not scanned for viruses."

Determine whether each of the functions $\log \left(n+1\right)$and $\log \left(n^2+1\right)$is $O\left(\log n\right)$.