Heb 9: 13

(KJV)
For if the blood of bulls and of goats, and the ashes of an heifer sprinkling the unclean, sanctifieth to the purifying of the flesh:
(NASB2020)
For if athe blood of goats and bulls, and bthe
I.e., ashes mixed in water
ashes of a heifer sprinkling those who have been defiled, sanctify for the
Lit purity
cleansing of the flesh,
(NET2full)
For if the blood of goats and bulls and the ashes of a young cow sprinkled on those who are defiled consecrated them and provided ritual purity,
tn Grk “for the purifying of the flesh.” The “flesh” here is symbolic of outward or ritual purity in contrast to inner purity, that of the conscience (cf. Heb 9:9).

Heb 9: 14

(KJV)
How much more shall the blood of Christ, who through the eternal Spirit offered himself without spot to God, purge your conscience from dead works to serve the living God?
(NASB2020)
how much more will fthe blood of Christ, who through
Or His eternal spirit
hthe eternal Spirit ioffered Himself without blemish to God, jcleanse your conscience from kdead works to serve lthe living God?

(NET2full)
how much more will the blood of Christ, who through the eternal Spirit offered himself without blemish to God, purify our
tc The reading adopted by the translation is attested by many authorities (A D* K P 365 1739* al). But many others (א D2 0278 33 1739c 1881 Maj lat sa) read “your” instead of “our.” The diversity of evidence makes this a difficult case to decide from external evidence alone. The first and second person pronouns differ by only one letter in Greek, as in English, also making this problem difficult to decide based on internal evidence and transcriptional probability. In the context, the author’s description of sacrificial activities seems to invite the reader to compare his own possible participation in OT liturgy as over against the completed work of Christ, so the second person pronoun “your” might make more sense. On the other hand, TCGNT 599 argues that “our” is preferable because the author of Hebrews uses direct address (i.e., the second person) only in the hortatory sections. What is more, the author seems to prefer the first person in explanatory remarks or when giving the logical grounds for an assertion (cf. Heb 4:15; 7:14). It is hard to reach a definitive conclusion in this case, but the data lean slightly in favor of the first person pronoun.
consciences from dead works to worship the living God.

Heb 9: 15

(KJV)
And for this cause he is the mediator of the new testament, that by means of death, for the redemption of the transgressions that were under the first testament, they which are called might receive the promise of eternal inheritance.
(NASB2020)
For this reason nHe is the omediator of a pnew covenant, so that, since a death has taken place for the redemption of the violations that were committed under the first covenant, those who have been qcalled may rreceive the promise of sthe eternal inheritance.
(NET2full)
And so he is the mediator
tn The Greek word μεσίτης (mesitēs, “mediator”) in this context does not imply that Jesus was a mediator in the contemporary sense of the word, i.e., he worked for compromise between opposing parties. Here the term describes his function as the one who was used by God to enact a new covenant which established a new relationship between God and his people, but entirely on God’s terms.
of a new covenant, so that those who are called may receive the eternal inheritance he has promised,
tn Grk “the promise of the eternal inheritance.”
since he died
tn Grk “a death having occurred.”
to set them free from the violations committed under the first covenant.
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